Metamodel-Based Optimization Method for Traffic Network Signal Design under Stochastic Demand

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Introduction
Trafc network design is a basic means to improve trafc fow distribution and alleviate trafc congestion in urban trafc networks.Te network design problem (NDP) is usually to determine optimal network supply decisions, such as adding new links or improving the capacity of existing ones, with certain objectives (e.g., maximizes social benefts or minimizes total travel cost), while considering users' route choice behavior [1][2][3][4][5][6].Tere exist diferent network supply decisions for the trafc network design problems, including road network expansion design, signal control optimization design, and road tolling design.For network trafc signal control, it focuses on determining optimal signal timing plans that can trigger better equilibrium fow patterns with optimal network performance.It is also called the combined trafc assignment and signal control problem [7][8][9][10][11][12] or anticipatory network trafc control [13][14][15][16] because the signal control anticipates the efect of route choice response.
Te trafc network signal design problem has been extensively explored in the literature.Allsop [17] frst proposed the concept of combined signal control and trafc assignment and developed an iterative optimization method to achieve the equilibrium solution by alternately modifying the signal timings and the equilibrium fows.However, it is reported that the solution of the iterative optimization method highly depends on the initial point (initial assignment), and the equilibrium solution is generally not necessarily the optimal solution [18][19][20].In view of the drawbacks of the iterative approach, Yang and Yagar [21] established the network signal control model from the perspective of global optimization and proposed a global optimization approach, which is usually formulated as a bilevel programming optimization model.Te upper level problem is the signal control optimization problem, which optimizes the network performance with fow constraints.Te lower level problem solves the user equilibrium (UE) problem [22][23][24][25] under the given signal timing plan.Te global optimization approach needs to simultaneously consider the trafc fow equilibrium and signal control optimization, which makes it time-consuming and difcult to solve.Te computational budget increases especially for large-scale road network problems.
Traditional trafc network design problems usually assume fxed or deterministic trafc conditions, such as fxed trafc demand.However, the transportation system is generally afected by many uncertain factors on demand and supply, for instance, OD demand fuctuations, link capacity variations, special events, and random route choice behavior.Ignoring the uncertainty efects in the decisionmaking process may result in inaccurate evaluations and suboptimal control plans [26][27][28].Li et al. [29] dealt with NDP under stochastic demand and reported that demand stochasticity afects the reliability of the optimal solution and its real application.Lv and Liu [30] also showed that the stochastic features of trafc demand will signifcantly afect the optimal signal control settings as well as the associated equilibrium fow pattern of the transportation network, leading to suboptimal network performance.Tis paper focuses on network trafc control under stochastic demand.To account for the impact of demand stochasticity and ensure the reliability of the solution, it is required to calculate the equilibrium fows under a large number of random demand scenarios, which substantially adds to the computation complexity of the control optimization problem.Te high computational budget of the control method that addresses the demand uncertainty limits its real-time and large-scale network applications.
Metamodel (or surrogate model) is a common method to solve nonlinear problems with high computational budget.It typically makes use of simple analytical models, which are called metamodels to approximate the original timeexpensive analysis or models, so as to improve the overall computation efciency [31].In general, metamodels can be classifed in two types: physical metamodels and functional metamodels.Te physical metamodel usually develops problem-specifc model to approximate the original problem from frst principle.Teir functional form and parameters have a physical or structural interpretation.Osorio and Bierlaire [32] considered simulation-based optimization approach for trafc signal control and developed a simplifed analytical queueing network model to approximate the complex queue network in the simulation.To improve the accuracy of the physical metamodel, it is necessary to conduct a model parameter ftting.A conventional two-step approach was usually applied to reduce the errors between the physical model and the real system [33].However, for complex transportation system, it is difcult to calibrate model parameters and establish an accurate physical model.
Another is the functional metamodel, which is composed of generic functions with general purposes.Te functional metamodel is usually developed based on analytical tractability, following a data-driven regression analysis approach.Hence, it does not include physical information regarding the underlying problem.A common way is to apply low-order polynomials for constructing the functional metamodel.In recent years, the functional metamodel has been gradually applied to the domain of trafc network design.Chen et al. [34] introduced Kriging surrogates to solve the network design problem under dynamic trafc assignment.Li et al. [35] proved the convergence of solving the continuous network design problem with the surrogate model and showed the advantages of the surrogate model in computation efciency.However, the functional metamodel relies highly on data, and the approximation performance outside the range of sample data is often unsatisfactory.It is typically that the data-driven method has a rigid requirement on the sample data and parameter ftting in order to achieve a good approximation performance.
To overcome the shortcomings of the physical and functional metamodels, Osorio and Bierlaire [32] proposed a metamodel that combines a functional component with a physical component to approximate the trafc queueing process.Te purpose of the functional component is to provide a more accurate local approximation, and the physical component is to provide a good global approximation.It has been shown that the combined metamodel method has a faster convergence speed and better ftting performance [36,37].Te above-mentioned studies focus on local intersection signal control, which does not consider the travelers' route choice behavior.Moreover, the demand uncertainty is not explicitly addressed.
Following the combined metamodeling approach, this paper proposes a metamodel-based optimization method for trafc network signal design under stochastic demand.Taking account of the stochastic features of trafc demand, a global optimization model is established with the goal of minimizing the expected total travel cost of the road network.Terefore, it needs to calculate the equilibrium fows under random demand scenarios and derive the expected performance.In order to improve the computational efciency of solving the average equilibrium trafc state, a metamodel that consists of a trafc assignment model (physical modeling) and a model bias (generic function) is constructed to approximate the expected equilibrium trafc fow.Tis paper further proposes to incorporate the gradient information of trafc fow with respect to the decision variable (the signal timing plan in our case) in the combined metamodel.By incorporating the gradient information, it is able to improve the parameter ftting performance and 2 Journal of Advanced Transportation hence the solution optimality.A gradient-based metamodel algorithm is then developed to solve the network signal control optimization problem.Te main contributions of this paper are summarized as follows: (1) A metamodel-based optimization method is developed for trafc network signal design under stochastic demand.To explicitly address the stochasticity in trafc demand and improve the computation efciency, a combined metamodel that consists of a physical modeling part and a model bias generic part is proposed to approximate the timeconsuming average equilibrium fow solution process.(2) A gradient metamodel scheme is further developed to make use of the gradient information of trafc fow to improve solution performance.(3) A gradient-based metamodel algorithm is proposed to solve the network signal control optimization problem.
Te rest of the paper is organized as follows.Section 2 elaborates the problem formulation and methodology of the metamodel-based optimization for trafc network signal design.Section 3 presents the numerical example on a test network.Insights into the properties of the proposed metamodel method and the solution performance of the method are demonstrated.Concluding remarks are discussed in Section 4.

Metamodel-Based Optimization Method for
Traffic Network Signal Design

Trafc Network Design Problems under Stochastic
Demand.In view of the inherent variations in trafc demand, in the trafc network design problem, the stochastic features of trafc demand need to be explicitly addressed in the optimization model to ensure reliable decisions.For the trafc network signal design problem under stochastic demand, it can be expressed as the problem of minimizing the expected total travel cost of the road network as follows: where Z i represents the travel cost of link i, which is a function of signal settings g (such as green splits) and link fow x.L represents the total number of links in the road network.Equation ( 1) is the objective function, i.e., minimizing the average of the total travel cost of all links.Constraint condition (2) represents the equilibrium fow constraint.Te equilibrium link fow pattern x is related to the signal settings g and the stochastic trafc demand d k , k � 1, 2, ..., K, represents the sample size of stochastic demand.Te equilibrium fow is derived by the trafc assignment model x Eq (g, d k ).Equation ( 3) is the signal timing constraint.Equation (4) sets the upper and lower limits of the signal control variables.According to the discussion above, in the presence of demand uncertainty, it is necessary to calculate the equilibrium link fow under a certain demand distribution.In other words, calculations of the trafc assignment model and the total travel cost function are repeated a large number of times, leading to a computational-intensive optimization problem.Terefore, the computational budget restricts the application of the stochastic network design in real-time or large-scale problems.

Metamodel-Based Optimization Method for Network
Signal Control.In order to improve the efciency of calculation, this paper proposes a metamodeling approach.As shown in equation ( 1), the objective is to minimize the expected total travel cost, which requires calculating the equilibrium fow under diferent demand scenarios.It usually involves a large number of scenarios (sample size) in order to achieve a comparable accuracy level, leading to a time-expensive calculation process.Terefore, the metamodel is developed as a surrogate of the expensive calculation process to improve computational efciency.First, we assume that the expected total travel cost is associated with the expected equilibrium link fow x ave � E[x Eq (g, d k )] under stochastic demand.In general, calculating the expected equilibrium fow takes most of the computation time.
To reduce the computation time, we introduce a metamodel x meta (g, d; β, θ) as a surrogate of x ave , to approximate the expensive calculation of the expected equilibrium fow with diferent demand scenarios.d is the average trafc demand.β and θ are parameters of the metamodel, whose feasible regions are Β and Θ, respectively.
Based on the metamodel, trafc network signal design problem under stochastic demand can be written as follows: where in equation ( 6) we calculate the expected equilibrium fow with the metamodel.Other constraints are the same of the original problem.In order to improve the approximation accuracy and make the approximate result of the metamodel closer to the actual average equilibrium fow, a suitable parameter set should be determined.Te parameter ftting can be formulated as a general least square error problem: Journal of Advanced Transportation where t is the iteration indicator and g t is signal settings at iteration t.
As discussed, the metamodel is an analytical approximation of the expensive calculation process of the original optimization, i.e., the calculation of average equilibrium fow under stochastic demand.Te metamodel-based optimization method then iterates over two main steps, including a metamodel ftting step and a signal control optimization step (i.e., the trafc network signal design).Figure 1 shows the interaction between diferent modules in the metamodelbased optimization framework.Te metamodel is constructed based on a sample of calculation results of the average equilibrium fow.Given the signal settings and stochastic demand, we can calculate the average equilibrium fow which involves solving the equilibrium fow for each demand and taking the average value.In the metamodel ftting step, based on the current sample of average equilibrium fow, the metamodel is ftted by solving the optimization problem (9).Ten, the signal control optimization step uses the ftted metamodel as constraint (6) to solve the signal control design problem and derive the optimal signal settings g t .Further, the updated signal settings are implemented in the expensive calculation process, which leads to a new calculation result of the average equilibrium fow x ave t .As the new sample becomes available, the metamodel is ftted again, leading to a more accurate metamodel.Te two steps iterate until convergence.At convergence, an accurate metamodel that approximates the original model can be obtained, and ultimately the optimal control scheme derived based on the metamodel should be close to the solution of the original trafc network design problem under stochastic demand.

Equilibrium Flow and Metamodel
Fitting.Te trafc network signal design considers the equilibrium fow constraint.From a network planning perspective, the signal control involves the interaction between the controller and travelers.Te controller anticipates travelers' route choice response when determining the signal settings, while travelers make route choice based on trafc conditions depending on the signal settings [13][14][15].Hence, the route choice response and the resulting equilibrium fow pattern, which is derived by solving a trafc assignment problem, are taken as constraints in the network signal design process.In general, fnding the solution of trafc assignment problem can be represented as a fxed-point problem.Te link fow determines the link travel cost, and the route travel cost calculated from the link travel cost will afect the route selection and hence the trafc fow assignment.Tis can be formulated as the following equations: where c is the link cost vector, which is calculated as a function of link fows x and signal settings g, h(c) represents the route fow, and B is incidence matrix of link-route fow, which can transform the route fow function into the link fow function F(c).Finding the solution of equations (10) represents a fxed-point problem, for which there exist diferent solution algorithms [38].Assuming that the link cost function C(x, g) is continuous and strictly increasing with x and the link fow function F(c) is continuous and monotonically decreasing with c, the existence and uniqueness of the fxed-point solution is guaranteed [39].Te solution of the fxed-point problem is the equilibrium fow.Te signal settings will afect the equilibrium state because the travel cost highly depends on the signal settings.Given a set of signal settings g 0 , the equilibrium fow can be expressed as follows: Te solution of this fxed-point problem depends on the link travel cost function and link fow function.Equation (11) shows that the equilibrium fow is related to the signal settings.
In this paper, the metamodel is used to approximate the average value of equilibrium fow under stochastic demand.As mentioned above, the metamodel that combines a functional component with a physical component is considered.Te purpose of the functional component is to provide a detailed local approximation and that of the physical component is to provide a good global approximation.Tis study develops a combined metamodel to approximate the average equilibrium fow.We formulate the trafc assignment model F(g; θ) as the physical modeling part.g is the set of signal settings, and θ is the set of model parameters to be calibrated.Te generic function is expressed as Φ(g; β).Ten the metamodel can be written as follows: x meta (g; β, θ) � F(g; θ) + Φ(g; β), (12) where β is the parameter of the generic function.In this paper, we consider the low-order polynomials function and defne Φ as follows: where N is the number of signal control variables.Terefore, the objective function of the metamodel ftting problem ( 9) can be written as follows: Te frst term is the error between the approximate result of the metamodel and the actual average equilibrium fow x ave t , and the second term is the ridge penalty term.Next, we elaborate on the development of the metamodel.First, the physical metamodel that only considers the simplifed problem-specifc model (the trafc assignment model in our case) is established.Ten, the concept of model bias is introduced, and a combined metamodel with the model bias as the generic part is proposed.To improve the solution performance, this paper further integrates the trafc fow gradient information into the combined metamodel.

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Journal of Advanced Transportation

Physical Metamodel.
Te physical metamodel only considers the simplifed physical model, that is, the trafc assignment model F(g; θ), as shown in equation (11).Generally, a two-step scheme is used to iteratively update model parameters θ and determine the optimal signal settings as follows: where t is the iteration indicator, and constraints of the optimization problem are not included for simplicity.Equation (15) represents the problem of model parameter estimation, which minimizes the distance between the approximate metamodel and the average equilibrium fow by updating the model parameters.Equation ( 16) represents the signal optimization problem.Based on the ftted metamodel, the optimal signal setting is calculated by minimizing the total travel cost, and these two steps iterate until convergence.

Combined Metamodel with Model Bias.
In view of the physical modeling error, this paper introduces a concept of model bias, which is defned as the error between the trafc assignment model and the average equilibrium link fow of the system as follows: At iteration step t, the model bias is calculated by the average equilibrium fow and the trafc assignment model with the corresponding signal settings g t as follows: With the help of model bias, a combined metamodel is developed as follows: Te combined metamodel consists of two terms.Te frst term is the trafc assignment model, which is the physical modeling part.Te second term of model bias b corresponds to the generic function part, which is updated by using the data during the iteration process.Te signal optimization design problem is formulated as follows: Te model bias b t is updated with the data obtained during the iteration process.

Gradient-Based Metamodel.
In this paper, a combined metamodel considering gradient information of trafc fow is proposed.In general, gradient is an important information for fnding the descending direction of the optimization problem.In the trafc assignment model, the gradient refects the variations of the trafc fow when the signal settings change.Incorporating gradient information generally improves solution performance in terms of convergence and solution optimality, i.e., faster convergence and better solution point [14].Patwary et al. [40] proposed a metamodel method with trafc fow gradient for an efcient calibration of large-scale trafc simulation models.For calculating the gradient of the equilibrium fow, this paper makes use of a fnite diference (FD) approach, which requires perturbing each signal control variable and calculates the corresponding derivative component.
where F(g) is the equilibrium fow function (i.e., the trafc assignment model) and h is a small perturbation on signal control variable g i .Calculating the gradient of the trafc assignment model, i.e., ∇F(g; θ), is trivial.Journal of Advanced Transportation gradient information, this paper applies a fnite diference approximation method [41], which uses the results recorded in previous iterations to estimate the Jacobian matrix of the average equilibrium fow.In each iteration, the Jacobian matrix is updated by the average equilibrium fow obtained in the historical iterations.Assuming that the number of control variables is n g , then n g + 1 control parameters and corresponding values of the average equilibrium fow are required, i.e., g t , . . ., g t−n g   and x ave t , . . ., x ave t−n g  .Te Jacobian matrix of iteration t can be calculated by the following formula: Terefore, the Jacobian matrix estimation based on the average equilibrium fow recorded in the historical iterations can be implemented as follows: Step 1: set a set of initial signal settings and the corresponding values of the average equilibrium fow under stochastic demand, i.e., g 0 , . . ., g 0−n g   and x ave 0 , . . .Step 2 and Step 3 are iterated until the convergence condition is satisfed.During the process, the gradient of the trafc assignment model ∇F(g; θ) and the gradient of the average equilibrium fow ∇x ave t should be calculated for each iteration step.Considering the gradient information of trafc fow, the following combined metamodel is constructed: ( Te gradient information is added to the calculation of model bias.As shown in equation (24), the model bias part is updated with x ave t − F(g t ; θ) + (∇x ave t − ∇F(g t ; θ))(g − g t ), corresponding to the generic function Φ(g; β) in equation (12).Now at iteration step t, the combined metamodel can be determined by the average equilibrium fow x ave t , the equilibrium fow calculated by trafc assignment model F(g t ; θ), the gradient of the average equilibrium fow ∇x ave t , and the gradient of trafc assignment model ∇F(g t ; θ).
Compared with the metamodel with model bias in equation ( 19), the gradient-based metamodel ( 24) not only considers the value of model bias but also takes account of the gradient information, i.e., the frst-order derivative information.Tis gradient-based metamodel method, which incorporates the gradient information of the metamodel at each local point g t , can ensure the frst-order optimality at convergence.Applying the gradient-based metamodel, the optimal signal setting g * t+1 at (t + 1) th iteration is determined by solving the following optimization problem: x meta (g; θ) ≥ 0, Choose a control step size μ and update the signal settings by where μ represents the control step size with a range of [0, 1].Algorithm 1 summarizes the solution process of the trafc network signal design problem under stochastic demand by using the combined metamodel method considering gradient information.

Numerical Examples
3.1.Simulation Setup.In this paper, a combined metamodel considering gradient information of trafc fow is proposed.Tis section establishes a simulation network to test the performance of the proposed method.Figure 2 is the test network, which consists of one OD pair (from node 1 to node 6), 8 links, and 5 routes.Link travel cost is calculated using a linearized Bureau of Public Roads (BPR) function.Te signal control plans of intersection node 3 and node 4 are decision variables.Assuming that the intersections operate in a two-phase timing plan, the green split is to be optimized.Te signal loss time is not considered in this case, i.e., g 2 + g 3 � 1 at intersection 3 and g 4 + g 6 � 1 at intersection 4.
Assuming that the travelers follow a nested logit (NL) structure for making route choice decisions, the probability of choosing route i can be expressed as follows: where the route travel cost is denoted by w.Te route choice set is divided into subsets J 1 , . . ., J k .ζ is the ratio of dispersion parameters of the two-layer structure of NL, associated with the frst and second choice levels, respectively.

Journal of Advanced Transportation
Te link travel cost c is derived by a linearized BPR function [42].Defning the free-fow travel time c 0 , saturation fow s, and a coefcient α, the link travel cost is expressed as a function of the link fow x and signal settings g as follows: For nonsignalized links, signal settings g are equal to 1.
Te equilibrium link fow that can be obtained by solving the fxed-point problem depends on the link travel cost and link fow under a given trafc demand.Te above calculations need to be carried out many times under the stochastic trafc demand to obtain the average equilibrium fow and then calculate the average total travel cost of the network.
In this paper, the metamodel method is introduced to simplify the trafc assignment calculation process and approximate the average equilibrium fow.In general, we cannot derive an accurate model of route choice behavior.In this case study, we assume that a multinomial logit (MNL) model with the dispersion parameter θ is used to describe the route choice and construct the metamodel of average equilibrium fow.Te probability is calculated by the model as follows: Te link travel cost is also represented by the BPR function (31).Te equilibrium fow is derived by solving the fxed-point problem with MNL and BPR function, which is used as the physical modeling part in the combined metamodel to approximate the average equilibrium fow.Te total travel time z is formulated as a function of the equilibrium fow and signal setting: Signal control decisions are to be made based on the metamodel, and the objective is to minimize the expected total travel cost on this network.All optimization problems in this numerical example are solved using the Python optimization toolbox.Characteristics of the network and model parameters are listed in Table 1.

Sensitivity Analysis of the Model Parameter.
As discussed, the trafc assignment model is used as the physical modeling part in the metamodel.In order to evaluate the role of model parameters and examine whether the model performance is sensitive to the parameters, we frst conduct Step 1: initialization.Set the parameters θ of trafc assignment model F(g; θ).Set a set of initial signal settings g 0 , . . ., g 0−n g  .
Step 2: apply the initial signal setting.Based on the initial signal settings g 0 , . . ., g 0−n g  , calculate the average equilibrium fow x ave � E[x Eq (g, d t )], and get the corresponding x ave 0 , . . ., x ave 0−n g  ; calculate the gradient of the trafc assignment model and the gradient of the initial average equilibrium fow, construct the combined metamodel as equation ( 24), and apply it into equations ( 25)-( 28) to solve the signal control optimization problem, obtain the control g 1 , and update the iteration step t � 1.
Step 3: calculate the average equilibrium fow.Implement g t to derive x ave � E[x Eq (g, d t )] and update the set of signal settings and the corresponding average equilibrium fow, i.e., g t , . . ., g t−n g   and x ave t , . . ., x ave t−n g  .
Step 6: check termination.Stop if the termination condition is satisfed; otherwise, set t � t + 1 and go to Step 3. ALGORITHM 1: Gradient-based metamodel algorithm.Journal of Advanced Transportation a sensitivity analysis on the trafc assignment model with respect to diferent model parameters.
In general, the parameter α in the BPR function is an important factor; we take it as the parameter to be calibrated.Next, we analyze the impact of the route choice parameter θ and the saturated fow s.Fixing α � 0.15 and saturated fow s � 2000, we adjust the parameter θ with a step size of 0.01 in the range of [0.5, 1.5] and calculate the corresponding link fow and total travel cost based on the physical metamodel.Te solution of optimal signal control plan is also derived with the corresponding parameters.Similarly, fxing parameter α and parameter θ, we adjust the saturated fow s with a step size of 10 in the range of [1700-2400] and calculate the corresponding change rates.Figures 3 and 4 show the variation and the change rate of link fow, total travel time, and optimal signal scheme with the route selection parameters θ and saturated fow s, respectively.Te results show that both parameters can afect the calculation results of the physical metamodel.In particular, the parameter θ has a more signifcant efect on the results when it is greater than 1.2.In view of the magnitude of the parameters, both have a fair impact on the network fow and control scheme.Terefore, the BPR parameter α, route choice parameter θ, and saturation fow s are taken as the ftting parameters of the physical metamodel.

Result Analysis and Comparison.
In this section, we test the performance of the proposed gradient-based metamodel method and compare it with the general physical metamodel and the combined metamodel method with model bias.We set the stochastic OD demand with a mean value of 2,000 and a variance of 10 and select a sample size of 500.Under stochastic demand, the equilibrium fows under 500 demand samples are solved, and the corresponding average total travel cost is calculated.

Te Benchmark Optimal Solution.
We frst calculate the optimal solution of the computation intensive network signal design problem under stochastic demand.Note that this is just used as a benchmark to examine the proposed method.Te goal of our proposed metamodel-based method is to reduce the computation time while retaining the solution optimality.Te optimal signal control scheme, link fow, and total travel cost are calculated by solving the signal control design problem with the NL model under stochastic demand, and the results are listed in Table 2. Figure 5 shows the expected total travel cost surface.Deviation of these results involves a computation-intensive process to calculate the average equilibrium fow and expected total travel cost.Te main purpose of listing the optimal control scheme here is to provide a benchmark for the subsequent method validation.In this paper, we propose a metamodel to approximate the time-consuming process to reduce the computation time of the optimization problem and make the optimization result as close as possible to the optimal signal control scheme.

Solution Performance of the Metamodel Method.
To illustrate the performance of the proposed method, we compare three metamodel schemes, i.e., the proposed gradient-based metamodel method (GD), the combined metamodel with model bias (bias), and a traditional physical metamodel method (two-step).By comparing with the physical metamodel method, we test the value of adding a model bias generic part in the combined metamodel.Furthermore, by comparing the GD method and the bias method, we validate the role of gradient information in improving solution optimality.Select diferent initial control points and analyze the convergence performance of the three methods.Te initial points are (g 2 , g 3 , g 4 , g 6 ) � (0.3, 0.7, 0.73, 0.27), (0.8, 0.2, 0.8, 0.2), and (0.2, 0.8, 0.2, 0.8), respectively, and the control step size μ � 0.7.Figures 6 and 7 illustrate the convergence performance and the optimal solutions of three methods under diferent initial points.Te selection of initial points typically afects the convergence process of the algorithm.Te results show that, compared with the physical metamodel, the combined metamodel greatly improves the optimal solution (in terms of reducing the expected total travel cost) with the help of the model bias.Moreover, by explicitly incorporating the gradient information of trafc fow, the gradient-based metamodel method further improves the solution performance and Journal of Advanced Transportation converges to a smaller total travel cost (i.e., convergence to a lower contour in Figure 7), which is closer to the original optimal solution.

Analysis of the Computation Time and Solution
Optimality.Solving the network signal control problem under stochastic demand requires carrying out the fxedpoint problems multiple times to obtain the corresponding equilibrium fow and the expected total travel cost, leading to a computationally expensive process.Terefore, this paper proposes a gradient-based metamodel method to approximate the average equilibrium fow function, replacing the time-consuming part of the signal control design problem.In this regard, the metamodel method can be evaluated from two aspects, namely, computational efciency and solution optimality (i.e., whether the optimal solution derived from the metamodel method is close to the optimal solution of the original problem).Tables 3-5 list the results of three metamodel methods with diferent initial points, including the computation time and the optimal solution performance (the expected total travel cost).In this example, diferent initial points have little infuence on the optimal solutions.Te entire process of metamodel-based optimization includes metamodel ftting, solving the optimal control, and calculating the sample average equilibrium trafc fow.Te time for metamodel ftting and solving the optimal control problem with the metamodel methods is in total approximately 0.04 s.Te time to obtain the average equilibrium trafc is about 0.28 s.Terefore, in terms of computation efciency, the time to solve the average equilibrium fow problem accounts for approximately 85% of the total calculation time in the metamodel optimization method.Tis shows that the timeconsuming process in the iteration is the multiple runs of the trafc assignment model under stochastic demand, which in turn validates the need of a more efcient surrogate for the calculation of the average equilibrium fow.An improvement factor (defned as the ratio of the computation time of the benchmark optimal control scheme to the computation time of the metamodel method) is introduced to capture the improvement of the computation time.Te results show that although there is a small reduction in solution optimality, the metamodel methods can signifcantly reduce the computation time (the computation time is reduced by 4.84 to 13.47 times under diferent initial points).With the help of the model bias, the combined metamodel can better approximate the original optimal solution.As indicated in Journal of Advanced Transportation Tables 3-5, compared with the traditional physical metamodel method, the combined metamodel method with model bias improves the total travel cost.Moreover, by incorporating the gradient information, the gradient-based method further improves the optimal solution.Te numerical results show that the proposed gradient-based metamodel method can efectively improve the computation efciency while slightly increasing the total travel cost (i.e., 0.09%, 0.09%, and 0.06% under the three initial points).Te infuence of control step size on the gradient-based metamodel method is further analyzed.Te step size adjustment methods with diferent optimization descent  We select the initial point (0.45, 0.55, 0.5, and 0.5) and compare these step size update methods, as shown in Figure 8. Adam and RMSprop converge slowly at the beginning because they limit the update within a certain range, which however makes the convergence process more stable.Terefore, diferent control steps will also afect the convergence process of the gradient-based metamodel method.In the solution process, we should carefully select

Conclusion
Tis paper developed a metamodel-based optimization method for trafc network signal design under stochastic OD demand.Solving the network design problem considering uncertainty typically involves an expensive calculation process to derive the equilibrium fows with a certain demand distribution.Tis paper applied a metamodeling approach and used a metamodel as a surrogate of the expensive calculation process of the average equilibrium fow, so as to enhance the overall computational efciency.More specifcally, based on the concept of model bias, a combined metamodel was developed, which integrates a physical modeling part (i.e., the trafc assignment model) and a model bias generic function.In order to further improve the solution performance, i.e., convergence and solution optimality, of the metamodel-based optimization method, the gradient information of trafc fow was incorporated in the metamodel, which provides a better descent direction of searching for the optimal solution.We tested the proposed gradient-based metamodel method on an example network.Tree methods were compared, including our proposed gradient-based metamodel, the combined metamodel with model bias, and the physical metamodel.Te comparison was conducted to investigate the importance of incorporating a model bias generic part and the trafc fow gradient information in the combined metamodel.Numerical results showed that there is a trade-of between computation time and solution optimality.Although there is a reduction in solution optimality, the metamodel methods signifcantly reduce the computation time (by 4.84 to 13.47 times under diferent initial points).By incorporating the model bias, the combined metamodel is able to better approximate the original optimal solution.Moreover, incorporating the trafc fow gradient information in the search algorithm further improves the solution performance.Comparison results indicated that the proposed gradient-based metamodel method can efectively improve the computation time with a small increase of 0.09% in the expected total travel cost.
In this paper, we apply the linear model to construct the generic function part of the combined metamodel.In future study, more functional forms including higher-order functions can be explored to improve the ftting performance of the method.Moreover, methods that can handle a larger amount of data should be explored.In addition, this paper focuses on developing the methodology and we test the efectiveness of the proposed metamodel method on a small example network.Our further research work will consider applications on larger road networks, probably based on certain trafc simulation models.

Data Availability
Te numerical example data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.Journal of Advanced Transportation

Figure 3 :
Figure 3: Te impact of route choice parameter θ on (a) link fow, (b) total travel cost, and (c) optimal control.

Figure 4 :
Figure 4: Te impact of saturate fow parameter s on (a) link fow, (b) total travel cost, and (c) optimal control.
Tis is because for each changed signal control variable, derivatives of equilibrium fows under diferent demand scenarios are required, which involve repeatedly solving the trafc assignment model.Regarding the computational budget on calculating the t � ∇E[x Eq (g t , d k )].
∇x ave 0 ;then solve the metamodel-based optimization; and derive g 1 .Step 2: apply g t , calculate the average equilibrium fow, update the set of signal control settings and fows, i.e., x ave 0−n g  ; calculate the initial Jacobian matrix (zx ave /zg)| g 0 from equation (21), i.e.,

Table 1 :
Network characteristics and model parameters.

Table 2 :
Network signal design under stochastic demand: optimal signal settings, link fows, and expected total travel cost.
the initial point and the control step under a specifc problem setting.