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- Publisher
- Hindawi Publishing Corporation
- ISSN
- 0197-6729
- eISSN
- 2042-3195
- DOI
- 10.1155/2023/3917657
- Publisher site
- See Article on Publisher Site
Abstract
Hindawi Journal of Advanced Transportation Volume 2023, Article ID 3917657, 15 pages https://doi.org/10.1155/2023/3917657 Research Article Metamodel-Based Optimization Method for Traffic Network Signal Design under Stochastic Demand 1 1 1 2 Wei Huang , Xuanyu Zhang, Haofan Cheng, and Jiemin Xie School of Intelligent Systems Engineering, Sun Yat-Sen University, Guangzhou 510006, China School of Intelligent Systems Engineering, Shenzhen Campus of Sun Yat-Sen University, Guangdong 518107, China Correspondence should be addressed to Jiemin Xie; xiejm28@mail.sysu.edu.cn Received 13 March 2023; Revised 20 April 2023; Accepted 24 April 2023; Published 27 May 2023 Academic Editor: Seungjae Lee Copyright © 2023 Wei Huang et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Trafcnetworkdesignproblems(NDPs)playanimportantroleinurbanplanning.Sincethereexistuncertaintiesintherealurban trafc network, neglecting the uncertainty factors may lead to unreasonable decisions. Tis paper considers the transportation network signal design problem under stochastic origin-destination (OD) demand. In general, solving this stochastic problem requires a large amount of computational budget to calculate the equilibrium fow corresponding to a certain demand dis- tribution,whichlimitsitsrealapplications.Toreducethecomputationaltimeincalculatingtheequilibriumfowunderstochastic demand, this paper proposes a metamodel-based optimization method. First, a combined metamodel that integrates a physical modeling part and a model bias generic part is developed. Te metamodel is used to approximate the time-consuming average equilibrium fow solution process, hence to improve the computational efciency. To further improve the convergence and the solutionoptimalityperformanceofthemetamodel-basedoptimization,thegradientinformationoftrafcfowwithrespecttothe signal planisincorporated intheoptimization model.Agradient-basedmetamodelalgorithmis thenproposed. Inthenumerical example, a six-node test network is used to examine the proposed metamodel-based optimization method. Te proposed combined metamodel is compared with the benchmark method to investigate the importance of incorporating a model bias generic part and the trafc fow gradient information in the combined metamodel. Although there is a reduction in solution optimality since the metamodel is an approximation of the original model, the metamodel methods greatly improve the computational efciency (the computational time is reduced by 4.84 to 13.47 times in the cases of diferent initial points). By incorporating the model bias, the combined metamodel can better approximate the original optimal solution. Moreover, in- corporating the gradient information of the trafc fow in the optimization search algorithm can further improve the solution performance. Numerical results show that the gradient-based metamodel method can efectively improve the computation efciency while slightly reducing the solution optimality (with an increase of 0.09% in the expected total travel cost). optimization design, and road tolling design. For network 1. Introduction trafc signal control, it focuses on determining optimal Trafc network design is a basic means to improve trafc signal timing plans that can trigger better equilibrium fow fow distribution and alleviate trafc congestion in urban patterns with optimal network performance. It is also called trafc networks. Te network design problem (NDP) is the combined trafc assignment and signal control problem usuallytodetermineoptimalnetworksupplydecisions,such [7–12] or anticipatory network trafc control [13–16] be- as adding new links or improving the capacity of existing cause the signal control anticipates the efect of route choice ones, with certain objectives (e.g., maximizes social benefts response. or minimizes total travel cost), while considering users’ Te trafc network signal design problem has been route choice behavior [1–6]. Tere exist diferent network extensively explored in the literature. Allsop [17] frst supply decisions for the trafc network design problems, proposed the concept of combined signal control and trafc including road network expansion design, signal control assignment and developed an iterative optimization method 2 Journal of Advanced Transportation complex queue network in the simulation. To improve the to achieve the equilibrium solution by alternately modifying the signal timings and the equilibrium fows. However, it is accuracy of the physical metamodel, it is necessary to conduct a model parameter ftting. A conventional two-step reported that the solution of the iterative optimization method highly depends on the initial point (initial assign- approach was usually applied to reduce the errors between ment), and the equilibrium solution is generally not nec- the physical model and the real system [33]. However, for essarily the optimal solution [18–20]. In view of the complex transportation system, it is difcult to calibrate drawbacks of the iterative approach, Yang and Yagar [21] model parameters and establish an accurate physical model. established the network signal control model from the Another is the functional metamodel, which is com- perspective of global optimization and proposed a global posed of generic functions with general purposes. Te functional metamodel is usually developed based on ana- optimization approach, which is usually formulated as a bilevel programming optimization model. Te upper level lytical tractability, following a data-driven regression anal- ysis approach. Hence, it does not include physical problem is the signal control optimization problem, which optimizes the network performance with fow constraints. information regarding the underlying problem. A common way is to apply low-order polynomials for constructing the Te lower level problem solves the user equilibrium (UE) problem [22–25] under the given signal timing plan. Te functional metamodel. In recent years, the functional global optimization approach needs to simultaneously metamodel has been gradually applied to the domain of consider the trafc fow equilibrium and signal control trafc network design. Chen et al. [34] introduced Kriging optimization, which makes it time-consuming and difcult surrogates to solve the network design problem under dy- to solve. Te computational budget increases especially for namic trafc assignment. Li et al. [35] proved the conver- large-scale road network problems. gence of solving the continuous network design problem with the surrogate model and showed the advantages of the Traditional trafc network design problems usually as- sume fxed or deterministic trafc conditions, such as fxed surrogate model in computation efciency. However, the functional metamodel relies highly on data, and the ap- trafc demand. However, the transportation system is generallyafectedbymanyuncertainfactorsondemandand proximationperformanceoutsidetherangeofsampledatais often unsatisfactory. It is typically that the data-driven supply, for instance, OD demand fuctuations, link capacity variations, special events, and random route choice be- method has a rigid requirement on the sample data and havior. Ignoring the uncertainty efects in the decision- parameter ftting in order to achieve a good approximation making process may result in inaccurate evaluations and performance. suboptimal control plans [26–28]. Li et al. [29] dealt with To overcome the shortcomings of the physical and NDP under stochastic demand and reported that demand functional metamodels, Osorio and Bierlaire [32] proposed stochasticityafectsthereliabilityoftheoptimalsolutionand a metamodel that combines a functional component with a physical component to approximate the trafc queueing its real application. Lv and Liu [30] also showed that the stochastic features of trafc demand will signifcantly afect process. Te purpose of the functional component is to provide a more accurate local approximation, and the the optimal signal control settings as well as the associated equilibrium fow pattern of the transportation network, physical component is to provide a good global approxi- leading to suboptimal network performance. Tis paper mation. It has been shown that the combined metamodel focuses on network trafc control under stochastic demand. method has a faster convergence speed and better ftting To account for the impact of demand stochasticity and performance[36,37].Teabove-mentionedstudiesfocuson ensure the reliability of the solution, it is required to cal- localintersectionsignalcontrol,whichdoesnotconsiderthe culate the equilibrium fows under a large number of ran- travelers’ route choice behavior. Moreover, the demand dom demand scenarios, which substantially adds to the uncertainty is not explicitly addressed. computation complexity of the control optimization prob- Following the combined metamodeling approach, this paperproposesametamodel-basedoptimizationmethodfor lem. Te high computational budget of the control method that addresses the demand uncertainty limits its real-time trafc network signal design under stochastic demand. Taking account of the stochastic features of trafc demand, and large-scale network applications. Metamodel(orsurrogatemodel)isacommonmethodto a global optimization model is established with the goal of solvenonlinearproblemswithhighcomputationalbudget.It minimizing the expected total travel cost of the road net- typically makes use of simple analytical models, which are work. Terefore, it needs to calculate the equilibrium fows called metamodels to approximate the original time- under random demand scenarios and derive the expected expensive analysis or models, so as to improve the overall performance. In order to improve the computational ef- computation efciency [31]. In general, metamodels can be ciency of solving the average equilibrium trafc state, a metamodel that consists of a trafc assignment model classifed in two types: physical metamodels and functional metamodels. Te physical metamodel usually develops (physical modeling) and a model bias (generic function) is constructed to approximate the expected equilibrium trafc problem-specifcmodeltoapproximatetheoriginalproblem from frst principle. Teir functional form and parameters fow.Tispaperfurtherproposestoincorporatethegradient information of trafc fow with respect to the decision have a physical or structural interpretation. Osorio and Bierlaire [32] considered simulation-based optimization variable(thesignal timingplaninourcase) inthecombined approachfortrafcsignalcontrolanddevelopedasimplifed metamodel. By incorporating the gradient information, it is analytical queueing network model to approximate the able to improve the parameter ftting performance and Journal of Advanced Transportation 3 hence the solution optimality. A gradient-based metamodel k � 1,2, ..., K, represents the sample size of stochastic de- algorithm is then developed to solve the network signal mand. Te equilibrium fow is derived by the trafc as- Eq control optimization problem. Te main contributions of signment model x (g, d ). Equation (3) is the signal timing this paper are summarized as follows: constraint. Equation (4) sets the upper and lower limits of the signal control variables. According to the discussion (1) A metamodel-based optimization method is de- above,inthepresenceofdemanduncertainty,itis necessary veloped for trafc network signal design under to calculate the equilibrium link fow under a certain de- stochastic demand. To explicitly address the sto- mand distribution. In other words, calculations of the trafc chasticity in trafc demand and improve the com- assignment model and the total travel cost function are putation efciency, a combined metamodel that repeated a large number of times, leading to a computa- consistsofaphysicalmodelingpartandamodelbias tional-intensive optimization problem. Terefore, the generic part is proposed to approximate the time- computational budget restricts the application of the sto- consuming average equilibrium fow solution chastic network design in real-time or large-scale problems. process. (2) A gradient metamodel scheme is further developed 2.2. Metamodel-Based Optimization Method for Network to make use of the gradient information of trafc fow to improve solution performance. Signal Control. In order to improve the efciency of cal- culation, this paper proposes a metamodeling approach. As (3) A gradient-based metamodel algorithm is proposed shown in equation (1), the objective is to minimize the to solve the network signal control optimization expected total travel cost, which requires calculating the problem. equilibrium fow under diferent demand scenarios. It Te rest of the paper is organized as follows. Section 2 usually involves a large number of scenarios (sample size) in elaborates the problem formulation and methodology of the order to achieve a comparable accuracy level, leading to metamodel-based optimization for trafc network signal a time-expensive calculation process. Terefore, the meta- design. Section 3 presents the numerical example on a test model is developed as a surrogate of the expensive calcu- network. Insights into the properties of the proposed lationprocesstoimprovecomputationalefciency.First,we metamodel method and the solution performance of the assume that the expected total travel cost is associated with ave Eq method are demonstrated. Concluding remarks are dis- the expected equilibrium link fow x � E[x (g, d )] cussed in Section 4. under stochastic demand. In general, calculating the ex- pectedequilibriumfowtakesmostofthecomputationtime. To reduce the computation time, we introduce a meta- 2. Metamodel-Based Optimization Method for meta ave model x (g, d; β, θ) as a surrogate of x , to approximate Traffic Network Signal Design the expensive calculation of the expected equilibrium fow with diferent demand scenarios. d is the average trafc 2.1. Trafc Network Design Problems under Stochastic demand. β and θ are parameters of the metamodel, whose Demand. In view of the inherent variations in trafc de- feasible regions are Β and Θ, respectively. mand, in the trafc network design problem, the stochastic features of trafc demand need to be explicitly addressed in Based on the metamodel, trafc network signal design problem under stochastic demand can be written as follows: the optimization model to ensure reliable decisions. For the trafc network signal design problem under stochastic de- min Z (g, x), mand, it can be expressed as the problem of minimizing the (5) i∈L expected total travel cost of the road network as follows: meta s.t. x � x (g, d; β, θ), (6) ⎡ ⎣ ⎤ ⎦ min E Z (g, x) , (1) i∈L f(g) � 0, (7) Eq (2) s.t. x � x g, d k � 1,2, . . . , K, g ≤ g ≤ g , (8) min max f(g) � 0, (3) where in equation (6) we calculate the expected equilibrium fow with the metamodel. Other constraints are the same of g ≤ g ≤ g , (4) min max theoriginalproblem.Inordertoimprovetheapproximation accuracyandmaketheapproximateresultofthemetamodel where Z represents the travel cost of link i, which is closer to the actual average equilibrium fow, a suitable a function of signal settings g (such as green splits) and link parameter set should be determined. Te parameter ftting fow x. L represents the total number of links in the road can be formulated as a general least square error problem: network. Equation (1) is the objective function, i.e., mini- mizing the average of the total travel cost of all links. Eq meta min f(α, β, θ) � min E x g , d − x g , d; β, θ , t k t Constraint condition (2) represents the equilibrium fow β,θ β,θ constraint. Te equilibrium link fow pattern x is related to (9) the signal settings g and the stochastic trafc demand d , k 4 Journal of Advanced Transportation where t is the iteration indicator and g is signal settings at routefow,whichcantransformtheroutefowfunctioninto iteration t. the link fow function F(c). Finding the solution of equa- tions (10) represents a fxed-point problem, for which there As discussed, the metamodel is an analytical approxi- mation of the expensive calculation process of the original exist diferent solution algorithms [38]. Assuming that the optimization,i.e.,thecalculationofaverageequilibriumfow link cost function C(x, g) is continuous and strictly in- under stochastic demand. Te metamodel-based optimiza- creasingwith xandthelinkfowfunction F(c)iscontinuous tion method then iterates over two main steps, including and monotonically decreasing with c, the existence and a metamodel ftting step and a signal control optimization uniqueness of the fxed-point solution is guaranteed [39]. step (i.e., the trafc network signal design). Figure 1 shows Te solution of the fxed-point problem is the equilibrium theinteractionbetweendiferentmodulesinthemetamodel- fow. Te signal settings will afect the equilibrium state based optimization framework. Te metamodel is con- because the travel cost highly depends on the signal settings. structed based on a sample of calculation results of the Given a set of signal settings g , the equilibrium fow can be average equilibrium fow. Given the signal settings and expressed as follows: stochastic demand, we can calculate the average equilibrium x � F C x, g � g . (11) fow which involves solving the equilibrium fow for each demand and taking the average value. In the metamodel Te solution of this fxed-point problem depends on the ftting step, based on the current sample of average equi- link travel cost function and link fow function. Equation librium fow, the metamodel is ftted by solving the opti- (11) shows that the equilibrium fow is related to the signal mization problem (9). Ten, the signal control optimization settings. step uses the ftted metamodel as constraint (6) to solve the In this paper, the metamodel is used to approximate the signal control design problem and derive the optimal signal average value of equilibrium fow under stochastic demand. settings g . Further, the updated signal settings are imple- As mentioned above, the metamodel that combines mented in the expensive calculation process, which leads to a functional component with a physical component is ave a new calculation result of the average equilibrium fow x . considered. Te purpose of the functional component is to As the new sample becomes available, the metamodel is provide a detailed local approximation and that of the ftted again, leading to a more accurate metamodel. Te two physical component is to provide a good global approxi- steps iterate until convergence. At convergence, an accurate mation. Tis study develops a combined metamodel to metamodel that approximates the original model can be approximatetheaverageequilibriumfow.Weformulatethe obtained,andultimatelytheoptimalcontrolschemederived trafc assignment model F(g; θ) as the physical modeling based on the metamodel should be close to the solution of part. g is the set of signal settings, and θ is the set of model the original trafc network design problem under stochastic parameters to be calibrated. Te generic function is demand. expressed as Φ(g; β). Ten the metamodel can be written as follows: 2.3. Equilibrium Flow and Metamodel Fitting. Te trafc meta x (12) (g; β, θ) � F(g; θ) + Φ(g; β), network signal design considers the equilibrium fow con- straint. From a network planning perspective, the signal where β is the parameter of the generic function. In this control involves the interaction between the controller and paper, we consider the low-order polynomials function and travelers. Te controller anticipates travelers’ route choice defne Φ as follows: response when determining the signal settings, while trav- elers make route choice based on trafc conditions (13) Φ(g; β) � β + β g , 0 i i i�1 depending on the signal settings [13–15]. Hence, the route whereNisthenumberofsignalcontrolvariables.Terefore, choice response and the resulting equilibrium fow pattern, the objective function of the metamodel ftting problem (9) whichisderivedbysolvingatrafcassignmentproblem,are can be written as follows: taken as constraints in the network signal design process. In general, fnding the solution of trafc assignment problem 2 N ave meta 2 min x − x g ; β, θ + β . t (14) t i can be represented as a fxed-point problem. Te link fow i�1 β,θ determines the link travel cost, and the route travel cost calculated from the link travel cost will afect the route Tefrsttermistheerrorbetweentheapproximateresult selection and hence the trafc fow assignment. Tis can be of the metamodel and the actual average equilibrium fow ave formulated as the following equations: x , and the second term is the ridge penalty term. Next, we elaborate on the development of the meta- c � C(x, g), model. First, thephysical metamodel that onlyconsiders the x � F(c) (10) simplifed problem-specifc model (the trafc assignment modelinourcase)isestablished.Ten,theconceptofmodel � Bh(c), bias is introduced, and a combined metamodel with the where c is the link cost vector, which is calculated as model bias as the generic part is proposed. To improve the solutionperformance,thispaperfurtherintegratesthetrafc a function of link fows x and signal settings g, h(c) rep- resents the route fow, and B is incidence matrix of link- fow gradient information into the combined metamodel. Journal of Advanced Transportation 5 Metamodel-based optimization Metamdoel fitting Traffic network signal design Metamdoel parameter Historical data meta Metamodel ave ave x (g,d;β,θ) {x ,..., x } β ,θ 1 t t t Optimization formulation ave Signal settings g Average equilibrium flow x t The expensive calculation process: t calculate the average equilibrium flow Figure 1: Metamodel-based optimization framework. 2.3.1. Physical Metamodel. Te physical metamodel only to the generic function part, which is updated by using the considers the simplifed physical model, that is, the trafc data during the iteration process. Te signal optimization assignment model F(g; θ), as shown in equation (11). design problem is formulated as follows: Generally, a two-step scheme is used to iteratively update meta x g ; θ � F g ; θ + b , (20) t t t model parameters θ and determine the optimal signal set- tings as follows: meta � � g � argmin zg, x g ; θ . t+1 t (21) � � ave � � g θ � argmin x − F g ; θ , � � (15) t t t Te model bias b is updated with the data obtained � argmin z g, F g; θ during the iteration process. g , t+1 t (16) where t is the iteration indicator, and constraints of the 2.3.3. Gradient-Based Metamodel. Inthispaper,acombined optimization problem are not included for simplicity. metamodel considering gradient information of trafc fow Equation (15) represents the problem of model parameter isproposed.Ingeneral,gradientisanimportantinformation estimation, which minimizes the distance between the ap- for fnding the descending direction of the optimization proximate metamodel and the average equilibrium fow by problem. In the trafc assignment model, the gradient re- updatingthemodelparameters.Equation(16)representsthe fects the variations of the trafc fow when the signal set- signaloptimizationproblem.Basedonthefttedmetamodel, tings change. Incorporating gradient information generally the optimal signal setting is calculated by minimizing the improvessolutionperformanceintermsofconvergenceand total travel cost, and these two steps iterate until solution optimality, i.e., faster convergence and better so- convergence. lution 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 2.3.2. Combined Metamodel with Model Bias. In view of the gradient of the equilibrium fow, this paper makes use of physical modeling error, this paper introduces a concept of a fnite diference (FD) approach, which requires perturbing model bias, which is defned as the error between the trafc eachsignalcontrolvariableandcalculatesthecorresponding assignment model and the average equilibrium link fow of derivative component. the system as follows: zF(g) F g , g , . . . , g + h, . . . , g − F g , g , . . . , g , . . . , g ave 1 2 i N 1 2 i N b � x − F(g; θ). (17) � , zg h At iteration step t, the model bias is calculated by the (22) average equilibrium fow and the trafc assignment model where F(g) is the equilibrium fow function (i.e., the trafc with the corresponding signal settings g as follows: assignment model) and h is a small perturbation on signal ave b � x g − F g ; θ . (18) control variable g . Calculating the gradient of the trafc t t t t assignment model, i.e., ∇F(g; θ), is trivial. However, it is With the help of model bias, a combined metamodel is computationally intensive to estimate the gradient of the developed as follows: actual average equilibrium fow, i.e., ave Eq meta ∇x � ∇E[x (g , d )]. Tis is because for each changed x (g; θ) � F(g; θ) + b. (19) t t k signal control variable, derivatives of equilibrium fows Tecombinedmetamodelconsistsoftwoterms.Tefrst under diferent demand scenarios are required, which in- term is the trafc assignment model, which is the physical volve repeatedly solving the trafc assignment model. Re- modelingpart.Tesecondtermofmodelbias bcorresponds garding the computational budget on calculating the 6 Journal of Advanced Transportation gradient information, this paper applies a fnite diference and the gradient of trafc assignment model ∇F(g ; θ). approximation method[41],which usestheresults recorded Compared with the metamodel with model bias in equation in previous iterations to estimate the Jacobian matrix of the (19), the gradient-based metamodel (24) not only considers average equilibrium fow. In each iteration, the Jacobian thevalueofmodelbiasbutalsotakesaccountofthegradient matrix is updated by the average equilibrium fow obtained information, i.e., the frst-order derivative information. Tis in the historical iterations. Assuming that the number of gradient-based metamodel method, which incorporates the control variables is n , then n +1 control parameters and gradient information of the metamodel at each local point g g corresponding values of the average equilibrium fow are g , can ensure the frst-order optimality at convergence. Applying the gradient-based metamodel, the optimal ave ave required, i.e., g , . . . , g and x , . . . , x . Te Ja- t t−n t t−n ∗ g g signal setting g at (t + 1) th iteration is determined by t+1 cobian matrix of iteration t can be calculated by the fol- solving the following optimization problem: lowing formula: ∗ meta T g � argmin z g, x (g; θ) , ave ave t+1 (25) x − x g t t−1 ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ave ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ zx ⎢ ⎥ −1⎢ ⎥ ⎢ ⎥ meta ave ⎢ ⎥ . . . ⎥ ⎢ ⎥, (23) � ∆g ⎢ ⎥ s.t. x (g; θ) � F(g; θ) + x − F g ; θ ⎢ ⎥ t ⎢ ⎥ t ⎢ ⎥ t ⎢ ⎥ ⎢ ⎥ zg ⎢ ⎥ ⎢ ⎥ (26) ⎢ ⎥ g ⎢ ⎥ t ⎢ ⎥ ave ⎢ ⎥ ⎢ ⎥ ⎢ T ⎥ ⎣ ⎦ + ∇x − ∇F g ; θ g − g , t t ave ave t x − x t t− n meta x (g; θ) ≥0, (27) where ∆g � [g − g , . . . , g − g ] . t t t−1 t t− n Terefore, the Jacobian matrix estimation based on the g ≤ g ≤ g . (28) min max averageequilibriumfowrecordedinthehistoricaliterations can be implemented as follows: Choose a control step size μ and update the signal settings by Step 1: set a set of initial signal settings and the cor- responding values of the average equilibrium fow g � g + μ g − g , (29) t+1 t t+1 t under stochastic demand, i.e., g , . . . , g and 0 0−n where μrepresentsthecontrolstepsizewitharangeof[0,1]. ave ave x , . . . , x ; calculate the initial Jacobian matrix 0 0−n g Algorithm 1 summarizes the solution process of the ave ave (zx /zg)| from equation (21), i.e., ∇x ;then solve trafc network signal design problem under stochastic de- g 0 mand by using the combined metamodel method consid- the metamodel-based optimization; and derive g . ering gradient information. Step 2: apply g , calculate the average equilibrium fow, update the set of signal control settings and fows, i.e., 3. Numerical Examples ave ave g , ..., g and x , . . . , x , and calculate t t−n t t−n g g ave ave 3.1. Simulation Setup. In this paper, a combined metamodel ∇x � (zx /zg)| t g considering gradient information of trafc fow is proposed. ave Step 3: input ∇x to solve the metamodel-based op- Tis section establishes a simulation network to test the timization and update g t+1 performance of the proposed method. Figure 2 is the test Step 2 and Step 3 are iterated until the convergence network, which consists of one OD pair (from node 1 to condition is satisfed. During the process, the gradient node 6), 8 links, and 5 routes. Link travel cost is calculated of the trafc assignment model ∇F(g; θ) and the gra- using a linearized Bureau of Public Roads (BPR) function. ave dient of the average equilibrium fow ∇x should be Te signal control plans of intersection node 3 and node 4 calculated for each iteration step. Considering the are decision variables. Assuming that the intersections gradient information of trafc fow, the following operate in a two-phase timing plan, the green split is to be combined metamodel is constructed: optimized.Tesignallosstimeisnotconsideredinthiscase, meta ave i.e., g + g � 1 at intersection 3 and g + g � 1 at 2 3 4 6 x (g; θ) � F(g; θ) + x − F g ; θ t t intersection 4. (24) ave + ∇x − ∇F g ; θ g − g . t t t 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: Te gradient information is added to the calculation of modelbias.Asshowninequation(24),themodelbiaspartis − w ζ ζ Y ave ave i 1 2 k e e updated with x − F(g ; θ) + (∇x − ∇F(g ; θ))(g − g ), sim t t t t t p [i] � · , (30) − w ζ ζ Y corresponding to the generic function Φ(g; β) in j 1 2 h e e j∈J h equation (12). Now at iteration step t, the combined metamodel can be wheretheroutetravelcostisdenotedby w.Teroutechoice ave determined by the average equilibrium fow x , the set is divided into subsets J , . . . , J . ζ is the ratio of dis- 1 k equilibrium fow calculated by trafc assignment model persion parameters of the two-layer structure of NL, asso- ave F(g ; θ), the gradient of the average equilibrium fow ∇x , ciated with the frst and second choice levels, respectively. t t Journal of Advanced Transportation 7 Step 1: initialization. Set the parameters θ of trafc assignment model F(g; θ). Set a set of initial signal settings g , . . . , g . 0 0−n Step 2: apply the initial signal setting. Based on the initial signal settings g , . . . , g , calculate the average equilibrium fow 0 0−n ave Eq ave ave x � E[x (g, d )], and get the corresponding x , . . . , x ; calculate the gradient of the trafc assignment model and the t 0 0−n 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 , and update the iteration step t � 1. ave Eq Step3:calculatetheaverageequilibriumfow.Implement g toderive x � E[x (g, d )]andupdatethesetofsignalsettingsandthe t t ave ave corresponding average equilibrium fow, i.e., g , . . . , g and x , . . . , x . t t−n t t−n g g Step4:updatethegradient-basedmetamodel.Calculatethegradientofthemetamodel ∇F(g ; θ)accordingtoequation(22);calculate ave ave theJacobianmatrix (zx /zg)| basedonequation(23)toobtain ∇x ;updatethecombinedmetamodelatthecurrentiteration,i.e., g t meta ave ave x (g; θ) � F(g; θ) + x − F(g ; θ) + (∇x − ∇F(g ; θ))(g − g ). t t t t t Step5:updatethesignalsetting.Calculate g bysolvingthecontroloptimization(25)–(28)withtheupdatedmetamodelandupdate t+1 the signal setting g � g + μ(g − g ). t+1 t t t+1 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. Link4 2 4 Link7 Link1 Route 1: Link 1 → 4 → 7 Route 2: Link 1 → 3 → 5 → 6 → 7 1 6 Route 3: Link 1 → 3 → 5 → 8 Link3 Link6 Route 4: Link 2 → 5 → 6 → 7 Route 5: Link 2 → 5 → 8 Link2 Link8 3 5 Link5 Figure 2: Test network. − w θ Te link travel cost c is derived by a linearized BPR model p [i] � . function [42]. Defning the free-fow travel time c , satu- (32) 0 − w θ e j∈J ration fow s, and a coefcient α, the link travel cost is k expressed as a function of the link fow x and signal settings Te link travel cost is also represented by the BPR g as follows: function (31). Te equilibrium fow is derived by solving the fxed-point problem with MNL and BPR function, which is c � C(x, g) used as the physical modeling part in the combined meta- (31) x model to approximate the average equilibrium fow. Te � c + α . total travel time z is formulated as a function of the equi- gs librium fow and signal setting: For nonsignalized links, signal settings g are equal to 1. z � z(g, x) Teequilibriumlinkfowthatcanbeobtainedbysolving (33) the fxed-point problem depends on the link travel cost and � C(x, g) · x. link fow under a given trafc demand. Te above calcu- lations need to be carried out many times under the sto- Signal control decisions are to be made based on the chastictrafcdemandtoobtaintheaverageequilibriumfow metamodel, and the objective is to minimize the expected and then calculate the average total travel cost of the total travel cost on this network. All optimization problems network. in this numerical example are solved using the Python In this paper, the metamodel method is introduced to optimization toolbox. Characteristics of the network and simplify the trafc assignment calculation process and ap- model parameters are listed in Table 1. proximate the average equilibrium fow. In general, we cannotderiveanaccuratemodelofroutechoicebehavior.In this case study, we assume that a multinomial logit (MNL) 3.2. Sensitivity Analysis of the Model Parameter. As dis- modelwiththedispersionparameter θisusedtodescribethe cussed, the trafc assignment model is used as the physical route choice and construct the metamodel of average modeling part in the metamodel. In order to evaluate the equilibrium fow. Te probability is calculated by the model role of model parameters and examine whether the model as follows: performance is sensitive to the parameters, we frst conduct 8 Journal of Advanced Transportation Table 1: Network characteristics and model parameters. Parameters Te NL model Te MNL model OD demand (veh/h) 2000 (average) 2000 Parameter α in cost function 0.13 0.15 Saturation fow s (veh/h) 2000 2000 Link 1 0.1 Link 1 0.1 Link 2 0.2 Link 2 0.8 Link 3 0.05 Link 3 0.05 Link 4 0.4 Link 4 0.4 Free-fow travel time c (h) Link 5 0.15 Link 5 0.15 Link 6 0.1 Link 6 0.7 Link 7 0.2 Link 7 0.2 Link 8 0.4 Link 8 0.4 Parameter θ of MNL 1 Parameter ζ of NL 0.1 Parameter ζ of NL 1 a sensitivity analysis on the trafc assignment model with is to reduce the computation time while retaining the so- respect to diferent model parameters. lution optimality. Te optimal signal control scheme, link In general, the parameter α in the BPR function is an fow, andtotal travel cost are calculated bysolving the signal importantfactor;wetakeitastheparametertobecalibrated. control design problem with the NL model under stochastic Next, we analyze the impact of the route choice parameter θ demand, and the results are listed in Table 2. Figure 5 shows and the saturated fow s. Fixing α � 0.15 and saturated fow the expected total travel cost surface. Deviation of these resultsinvolvesacomputation-intensiveprocesstocalculate s � 2000,weadjusttheparameter θwithastepsizeof0.01in the range of [0.5,1.5] and calculate the corresponding link the average equilibrium fow and expected total travel cost. Temainpurposeoflistingtheoptimalcontrolschemehere fow and total travel cost based on the physical metamodel. Te solution of optimal signal control plan is also derived is to provide a benchmark for the subsequent method with the corresponding parameters. Similarly, fxing pa- validation. In this paper, we propose a metamodel to ap- rameter α and parameter θ, we adjust the saturated fow s proximate the time-consuming process to reduce the with a step size of 10 in the range of [1700–2400] and computationtimeoftheoptimizationproblemandmakethe calculate the corresponding change rates. Figures 3 and 4 optimization result as close as possible to the optimal signal show the variation and the change rate of link fow, total control scheme. travel time, and optimal signal scheme with the route se- lection parameters θ and saturated fow s, respectively. Te results show that both parameters can afect the calculation 3.3.2. Solution Performance of the Metamodel Method. results of the physical metamodel. In particular, the pa- To illustrate the performance of the proposed method, we rameter θ has a more signifcant efect on the results when it compare three metamodel schemes, i.e., the proposed is greater than 1.2. In view of the magnitude of the pa- gradient-based metamodel method (GD), the combined rameters, both have a fair impact on the network fow and metamodelwithmodelbias(bias),andatraditionalphysical control scheme. Terefore, the BPR parameter α, route metamodel method (two-step). By comparing with the choice parameter θ, and saturation fow s are taken as the physical metamodel method, we test the value of adding ftting parameters of the physical metamodel. 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 3.3. Result Analysis and Comparison. In this section, we test improvingsolutionoptimality.Selectdiferentinitialcontrol the performance of the proposed gradient-based metamodel pointsandanalyzetheconvergenceperformanceofthethree methodandcompareitwiththegeneralphysicalmetamodel methods. Te initial points are (g , g , g , g ) �(0.3, 0.7, 2 3 4 6 and the combined metamodel method with model bias. We 0.73, 0.27), (0.8, 0.2, 0.8, 0.2), and (0.2, 0.8, 0.2, 0.8), re- set the stochastic OD demand with a mean value of 2,000 spectively, and the control step size μ � 0.7. Figures 6 and 7 and a variance of 10 and select a sample size of 500. Under illustrate the convergence performance and the optimal stochasticdemand,theequilibriumfowsunder500demand solutionsofthreemethodsunderdiferentinitialpoints.Te samples are solved, and the corresponding average total selection of initial points typically afects the convergence travel cost is calculated. process of the algorithm. Te results show that, compared with the physical metamodel, the combined metamodel 3.3.1. Te Benchmark Optimal Solution. We frst calculate greatly improves the optimal solution (in terms of reducing the optimal solution of the computation intensive network theexpectedtotaltravelcost)withthehelpofthemodelbias. signal design problem under stochastic demand. Note that Moreover, by explicitly incorporating the gradient in- this is just used as a benchmark to examine the proposed formation of trafc fow, the gradient-based metamodel method.Tegoalofourproposedmetamodel-basedmethod method further improves the solution performance and Journal of Advanced Transportation 9 The change rate of link flow The change rate of total travel cost 1000 2900 10000 100000 950 100 900 2000 100 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Parameter θ Parameter θ Link Flow Total Travel Cost The change rate of link flow The change rate of total travel cost (a) (b) The change rate of optimal signal scheme 0.8 0.7 0.6 1 0.5 0.1 0.4 0.01 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Parameter θ Optimal Signal Scheme The change rate of optimal signal scheme (c) Figure 3: Te impact of route choice parameter θ on (a) link fow, (b) total travel cost, and (c) optimal control. converges to a smaller total travel cost (i.e., convergence to metamodel-based optimization includes metamodel ftting, a lower contour in Figure 7), which is closer to the original solving the optimal control, and calculating the sample optimal solution. average equilibrium trafc fow. Te time for metamodel ftting and solving the optimal control problem with the metamodel methods is in total approximately 0.04s. Te time to obtain the average equilibrium trafc is about 0.28s. 3.3.3. Analysis of the Computation Time and Solution Terefore, in terms of computation efciency, the time to Optimality. Solving the network signal control problem solve the average equilibrium fow problem accounts for under stochastic demand requires carrying out the fxed- approximately 85% of the total calculation time in the point problems multiple times to obtain the corresponding metamodel optimization method. Tis shows that the time- equilibriumfowandtheexpectedtotaltravelcost,leadingto consumingprocessintheiterationisthemultiplerunsofthe a computationally expensive process. Terefore, this paper trafc assignmentmodel understochastic demand,which in proposes a gradient-based metamodel method to approxi- turn validates the need of a more efcient surrogate for the mate the average equilibrium fow function, replacing the calculation of the average equilibrium fow. An improve- time-consuming part of the signal control design problem. ment factor (defned as the ratio of the computation time of Inthisregard,themetamodelmethodcanbeevaluatedfrom the benchmark optimal control scheme to the computation two aspects, namely, computational efciency and solution time of the metamodel method) is introduced to capture the optimality (i.e., whether the optimal solution derived from improvementofthecomputationtime.Teresultsshowthat themetamodelmethodisclosetotheoptimalsolutionofthe although there is a small reduction in solution optimality, original problem). the metamodel methods can signifcantly reduce the com- Tables 3–5 list the results of three metamodel methods putation time (the computation time is reduced by 4.84 to with diferent initial points, including the computation time 13.47 times under diferent initial points). With the help of and the optimal solution performance (the expected total the model bias, the combined metamodel can better ap- travelcost).Inthisexample,diferentinitialpointshavelittle proximate the original optimal solution. As indicated in infuence on the optimal solutions. Te entire process of Link Flow Optimal Signal Scheme Total Travel Cost 10 Journal of Advanced Transportation The change rate of link flow The change rate of total travel cost −1 −2 ×10 ×10 3000 6.0 1000 2.0 2750 4.5 975 1.5 2500 3.0 950 1.0 2250 1.5 925 0.5 2000 0.0 900 0.0 Parameter S Parameter S Total Travel Cost Link Flow The change rate of total travel cost The change rate of link flow (a) (b) The change rate of optimal signal scheme −4 ×10 0.55 1.0 0.52 0.8 0.49 0.6 0.46 0.4 0.43 0.2 0.40 0.0 Parameter S Optimal Signal Scheme The change rate of optimal signal scheme (c) Figure 4: Te impact of saturate fow parameter s on (a) link fow, (b) total travel cost, and (c) optimal control. Table 2: Network signal design under stochastic demand: optimal signal settings, link fows, and expected total travel cost. Optimal solution for signal design problem under stochastic demand Signal green split (g , g , g , g ) � (0.61,0.39,0.34,0.66) 2 3 4 6 Link fow (veh/h) Link 1 1022 Link 2 979.4 Link 3 652.9 Link 4 369.1 Link 5 1632.3 Link 6 814.5 Link 7 1183.7 Link 8 817.7 Total travel cost (veh∙h) 2008.76 Tables 3–5, compared with the traditional physical meta- metamodel method can efectively improve the computa- model method, the combined metamodel method with tion efciency while slightly increasing the total travel cost model bias improves the total travel cost. Moreover, by (i.e.,0.09%,0.09%,and0.06%underthethreeinitialpoints). incorporating the gradient information, the gradient-based Te infuence of control step size on the gradient-based method further improves the optimal solution. Te nu- metamodel method is further analyzed. Te step size ad- merical results show that the proposed gradient-based justment methods with diferent optimization descent Link Flow Optimal Signal Scheme 1800 2400 Total Travel Cost 2400 Journal of Advanced Transportation 11 Total travel cost 0.8 0.6 0.2 0.4 0.4 4 0.6 0.2 2 0.8 Figure 5: Te expected total travel cost surface. Total travel cost Total travel cost Total travel cost 2080 2080 2060 2060 2050 2050 2040 2040 2030 2030 2020 2020 2010 2010 2468 10 12 2 4 6 8 10 12 2468 10 12 Iteration Iteration Iteration two-step two-step two-step bias bias bias GD-I GD-I GD-I (a) (b) (c) Figure 6: Te convergence performance under diferent initial points: (a) (0.3, 0.7, 0.73, 0.27), (b) (0.8, 0.2, 0.8, 0.2), and (c) (0.2, 0.8, 0.2, 0.8). directions are considered. Te commonly used step size beginning because they limit the update within a certain update methods include Adam, Momentum, and RMSprop range, which however makes the convergence process more algorithms. We select the initial point (0.45, 0.55, 0.5, and stable. Terefore, diferent control steps will also afect the 0.5) and compare these step size update methods, as shown convergence process of the gradient-based metamodel in Figure 8. Adam and RMSprop converge slowly at the method. In the solution process, we should carefully select 12 Journal of Advanced Transportation Total Travel Time Total Travel Time 0.9 5880 0.9 5880 0.8 0.8 3500 3000 3000 2080 2080 2500 2500 0.7 0.7 2350 2350 2200 2200 0.6 0.6 2120 2030 2120 2080 2080 0.5 2120 0.5 2060 2060 2050 2050 2030 2350 0.4 0.4 2013 2040 2040 2020 2020 2015 2030 2030 0.3 0.3 2020 2020 2350 2011 2015 2015 0.2 0.2 2060 2013 2013 2060 2040 2011 2011 0.1 2008 0.1 2500 2008 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Signal green solit (g ) Signal green solit (g ) GD-I GD-I two-step two-step bias bias (a) (b) Total Travel Time 0.9 0.8 2200 0.7 0.6 2030 2120 0.5 0.4 0.3 2060 2013 0.2 2500 2008 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Signal green solit (g ) GD-I two-step bias (c) Figure 7: Te total travel cost under diferent initial points: (a) (0.3, 0.7, 0.73, 0.27), (b) (0.8, 0.2, 0.8, 0.2), and (c) (0.2, 0.8, 0.2, 0.8). Table 3: Computation time and optimal solution with initial points (0.3, 0.7, 0.73, and 0.27). Improvement Total Increase in total Method Computation time (s) factor (iteration) travel cost (veh/h) travel cost (%) Te original problem under stochastic demand 10.236 2008.76 Physical metamodel 0.8067 12.69 (5) 2052.96 2.20 Combined metamodel method with model bias 0.7598 13.47 (5) 2013.22 0.22 Gradient-based metamodel 1.8562 5.51 (10) 2010.59 0.09 Table 4: Computation time and optimal solution with initial points (0.8, 0.2, 0.8, and 0.2). Improvement Total Increase in total Method Computation time (s) factor (iteration) travel cost (veh/h) travel cost (%) Te original problem under stochastic demand 9.6482 2008.76 Physical metamodel 0.8985 10.73 (5) 2052.92 2.20 Combined metamodel method with model bias 0.7878 12.25 (5) 2013.18 0.22 Gradient-based metamodel 1.9916 4.84 (11) 2010.57 0.09 Signal green solit (g ) Signal green solit (g ) Signal green solit (g ) 4 Journal of Advanced Transportation 13 Table 5: Computation time and optimal solution with initial points (0.2, 0.8, 0.2, and 0.8). Improvement Total Increase in total Method Computation time (s) factor (iteration) travel cost (veh/h) travel cost (%) Te original problem under stochastic demand 9.8356 2008.76 Physical metamodel 0.8975 10.96 (5) 2052.52 2.18 Combined metamodel method with model bias 0.7639 12.88 (5) 2012.95 0.21 Gradient-based metamodel 1.6565 5.94 (9) 2009.94 0.06 Total travel cost Total travel cost 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 2 4 6 8 10 12 14 16 18 20 Signal green split g Iteration μ=0.7 Momentum Adam RMSprop Figure 8: Convergence performance under diferent step sizes. the initial point and the control step under a specifc a reduction in solution optimality, the metamodel methods problem setting. signifcantly reduce the computation time (by 4.84 to 13.47 times under diferent initial points). By incorporating 4. Conclusion the model bias, the combined metamodel is able to better approximate the original optimal solution. Moreover, in- Tis paper developed a metamodel-based optimization corporating the trafc fow gradient information in the method for trafc network signal design under stochastic search algorithm further improves the solution perfor- OD demand. Solving the network design problem consid- mance. Comparison results indicated that the proposed ering uncertainty typically involves an expensive calculation gradient-based metamodel method can efectively improve process to derive the equilibrium fows with a certain de- the computation time with a small increase of 0.09% in the mand distribution. Tis paper applied a metamodeling expected total travel cost. approach and used a metamodel as a surrogate of the ex- In this paper, we apply the linear model to construct the pensive calculation process of the average equilibrium fow, generic function part of the combined metamodel. In future so as to enhance the overall computational efciency. More study, more functional forms including higher-order specifcally, based on the concept of model bias, a combined functions can be explored to improve the ftting perfor- metamodel was developed, which integrates a physical mance of the method. Moreover, methods that can handle modeling part (i.e., the trafc assignment model) and a larger amount of data should be explored. In addition, this a model bias generic function. In order to further improve paper focuses on developing the methodology and we test the solution performance, i.e., convergence and solution the efectiveness of the proposed metamodel method on optimality, of the metamodel-based optimization method, a small example network. Our further research work will the gradient information of trafc fow was incorporated in consider applications on larger road networks, probably the metamodel, which provides a better descent direction of based on certain trafc simulation models. searching for the optimal solution. We tested the proposed gradient-based metamodel method on an example network. Data Availability Tree methods were compared, including our proposed gradient-based metamodel, the combined metamodel with Te numerical example data used to support the fndings of model bias, and the physical metamodel. Te comparison this study areavailable from thecorresponding author upon was conducted to investigate the importance of in- request. corporating a model bias generic part and the trafc fow gradient information in the combined metamodel. Nu- Conflicts of Interest merical results showed that there is a trade-of between computationtimeandsolutionoptimality.Althoughthereis Te authors declare that they have no conficts of interest. Signal green split g 4 14 Journal of Advanced Transportation Research Part C: Emerging Technologies, vol. 67, pp. 243–265, Acknowledgments [15] W. Huang, F. Viti, and C. M. J. 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Published: May 27, 2023