Calibrating Path Choices and Train Capacities for Urban Rail Transit Simulation Models Using Smart Card and Train Movement Data
Baichuan Mo, Zhenliang Ma, Haris N. Koutsopoulos, Jinhua Zhao
CCalibrating Path Choices and Train Capacities for Urban Rail TransitSimulation Models Using Smart Card and Train Movement Data
Baichuan Mo a , Zhenliang Ma b, ∗ , Haris N. Koutsopoulos c , Jinhua Zhao d a Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 b Department of Civil Engineering, Monash University, Melbourne, VIC 3800 c Department of Civil and Environmental Engineering, Northeastern University, Boston, MA 02115 d Department of Urban Studies and Planning, Massachusetts Institute of Technology, Cambridge, MA 20139
Abstract
Transit network simulation models are often used for performance and retrospective analysis of urban railsystems, taking advantage of the availability of extensive automated fare collection (AFC) and automatedvehicle location (AVL) data. Important inputs to such models, in addition to origin-destination flows, includepassenger path choices and train capacity. Train capacity, which has often been overlooked in the literature,is an important input that exhibits a lot of variabilities. The paper proposes a simulation-based optimization(SBO) framework to simultaneously calibrate path choices and train capacity for urban rail systems usingAFC and AVL data. The calibration is formulated as an optimization problem with a black-box objectivefunction. Seven algorithms from four branches of SBO solving methods are evaluated. The algorithmsare evaluated using an experimental design that includes five scenarios, representing different degrees ofpath choice randomness and crowding sensitivity. Data from the Hong Kong Mass Transit Railway (MTR)system is used as a case study. The data is used to generate synthetic observations used as “ground truth”.The results show that the response surface methods (particularly Constrained Optimization using ResponseSurfaces) have consistently good performance under all scenarios. The proposed approach drives large-scalesimulation applications for monitoring and planning.
Keywords:
Simulation-based optimization, path choice, train capacity, smart card data, urban railwaysystems
1. Introduction
Urban rail systems are important components of the urban transportation system. Given their highreliability and large capacity, they have attracted high passenger demand. However, high demand also leadsto problems such as overcrowding and disruptions, which decrease the level of service and impact passengers.To maintain service reliability and develop efficient response strategies, it is crucial for operators to betterunderstand passenger demand and flow patterns in the network.Transit network loading (or simulation) models for metro systems, powered by automated collected data,provide a useful instrument for network performance monitoring. They enable operators to characterize thelevel of service and make decisions accordingly. A typical network loading model requires Origin-Destination(OD) matrix, supply information, and path choice fractions as input. The supply information includesthe transit network topology, actual vehicle movement data, and vehicle capacity. Thanks to the widedeployment of automated fare collection (AFC) and automated vehicle location (AVL) systems, the ODdemand and train movement data can be directly obtained. However, obtaining the corresponding pathchoices and quantifying reasonable vehicle capacity remains a challenge. According to Liu et al. (2016)and Preston et al. (2017), train capacity, defined as the maximum train load when remaining passengers inthe platform denied boarding, may vary depending on the crowding levels in trains and on platforms, andpassenger attitudes. The calibration of path choices and train capacity can improve the accuracy of network ∗ Corresponding author: Zhenliang Ma. Emails: B. Mo, [email protected]; Z. Ma, [email protected].; H. Koutsopoulos,[email protected]; J. Zhao, [email protected]
Preprint submitted to test February 16, 2021 a r X i v : . [ c s . OH ] F e b oading models for performance monitoring. Thus, these models can provide better information to operatorsto adjust operating strategies, relieve congestion, and improve efficiency.Traditionally, path choices are inferred with data from on-site surveys that are used to estimate pathchoice models. However, surveys are time-consuming and labor-intensive, limiting their real-world usage. Toovercome these disadvantages, path choice estimation methods based on AFC data have been proposed inthe literature. AFC systems provide the exact locations and times of passengers’ entry and exit transactions,which can be used to extract OD demand and passengers’ journey times. They provide rich information foranalyzing passenger behavior (Koutsopoulos et al., 2019).In an urban rail system operated near its capacity, five critical parameters are correlated with eachother: OD demand, journey time, left behind (or denied boarding), path choices, and train capacity. Therelationship of these parameters can be explained in Figure 1. OD demand is the input and journey timeis the output (OD exit flow is a combination of the two), which can both be observed from the AFC data.Path choices, train capacity, and left behind are not observable in the AFC data. Journey time is directlyaffected by path choices and left behind (left behind can increase the waiting time). Left behind is directlyaffected by path choices and train capacity. This figure indicates the complexity of path choice estimationusing AFC data. The dependencies of different parameters (e.g., path choices v.s. train capacity) should becaptured. Figure 1: Relationship among critical parameters in urban rail systems
In the context of path choice estimation, the AFC data-based methods can be categorized into twogroups: path-identification methods (Kusakabe et al., 2010; Zhou and Xu, 2012; Kumar et al., 2018; Zhuet al., 2020) and parameter-inference methods (Sun and Xu, 2012; Sun et al., 2015; Zhao et al., 2017; Xuet al., 2018; Mo et al., 2020a). The former studies aim to identify the exact path chosen by each user andeven the train they boarded. Path attributes are used to evaluate how likely a path is chosen for a passenger’strip from their observed origins to their observed destinations. The latter studies formulate probabilisticmodels to describe passengers’ decision-making behavior. Bayesian inference is usually used to estimate thecorresponding parameters and thus derive the path choice fractions. Despite using different methods, thekey components for those AFC data-based studies are similar. They all attempt to match the model-derivedjourney times with the observed journey times from AFC data. However, many of these studies either assumea known constant train capacity or specify a known link-impedance function. As shown in Figure 1, journeytimes depend on both path choices and train capacity. An unreasonable setting of train capacity may causecalibration bias of path choices. Simultaneous calibration of both parameters is more reasonable.Train capacity is a vague concept. Normally trains may not reach their designed physical capacity forvarious reasons (e.g. passengers may decide not to board due to the crowding Liu et al. (2016)). Therefore,assuming a fixed physical capacity or fixed link-impedance function (in many previous studies) may not bea reasonable assumption in real-world situations. Only a few studies have explored the calibration of actualtrain capacity in the rail system. Liu et al. (2016) proposed the concept of “willingness to board” (WTB) todescribe the varied capacity in a bus system, and estimated passengers’ WTB using a least square method.Xu and Yong proposed a passenger boarding model which revealed that the number of actually boardingpassengers in a crowded train was closely related to the number of queuing passengers and train load. Moet al. (2020c) proposed an effective capacity model that recognized train capacity may vary across stationsdepending on the corresponding number of queuing passengers and train load. The calibration of traincapacity or WTB usually requires the AFC data with passengers’ boarding and journey time information.However, this information may also be affected by path choices, which were neglected in previous studies.2o fill these research gaps, we propose a simulation-based optimization (SBO) model to calibrate pathchoices and train capacity simultaneously and also explores the efficiency of typical SBO solution algo-rithms. The calibration problem is formulated as an optimization problem using AFC and AVL data. Theformulation can capture the interaction among these variables and their impact on journey times. Sevenoptimizers (solving algorithms) from four branches of SBO solving methods are implemented for compar-ative analysis. They include Generic Algorithm (GA), Simulated Annealing (SA), Nelder-Mead SimplexAlgorithm (NMSA), Mesh Adaptive Direct Search (MADS), Simultaneous Perturbation Stochastic Approx-imation (SPSA), Bayesian Optimization (BYO) and Constrained Optimization using Response Surfaces(CORS). We compare these SBO solving algorithms within a limited computational budget, defined by thenumber of function evaluations. Data from the Hong Kong Mass Transit Railway (MTR) system providethe foundation for a realistic case study. The major contribution of this paper is twofold: • Proposing an optimization model to simultaneously estimate path choices and train capacities usingAFC and AVL data. It addresses the typical assumption of fixed and known train capacities in existingpath choice estimation studies using smart card data. • Validating the model using a busy urban rail network and analyzing the performance of SBO solutionalgorithms using systematic experiments, representing different degrees of users’ randomness in pathchoice and their sensitivity to crowding.The remainder of the paper is organized as follows: In Section 2, we illustrate the SBO problem formu-lation. Section 3 briefly describes the various SBO methods used in this study. The proposed framework isused in a case study with data from the Hong Kong MTR network in Section 4. The results are used tocompare the performance of different algorithms. Section 5 concludes the paper by summarizing the mainfindings and discussing future research directions.
2. Methodology
The paper aims to calibrate simultaneously the train capacity and path choices using readily availabledata in the closed fare payment systems (require ticket validation at both tap-in and tap-out stations). Tocapture the interaction among different variables in Figure 1, we use a schedule-based network loading modelwith capacity constraints (described in Section 2.1). It outputs a list of performance metrics given a set ofinputs including OD demand, timetables/AVL, network, train capacity and path choices. The calibrationof path choice and capacity is formulated as an optimization model that attempts to minimize the errorbetween network loading model outputs (e.g., journey time, which is a function of path choices and traincapacity) and the corresponding quantities directly observed from the AFC data.
Transit network loading (TNL) models aim to assign passengers over a transit network given the (dy-namic) OD entry demand and path choices. In this study, we adopt an event-driven schedule-based TNLmodel proposed by Mo et al. (2020c). The model takes OD entry demand (number of tap-in passengers bytime), path choices, train arrival and departure times from stations, train capacity, and infrastructure infor-mation (e.g. network topology) as inputs, and outputs the passengers’ tap-out times, train loads, waitingtimes, and other network performance indicators of interest.Figure 2 illustrates the main functions of the TNL model Mo et al. (2020c). Three objects are defined:train, waiting queue (on the platform), and passengers. An event is defined as a train arrival at, or departurefrom, a station. Events are ordered chronologically. New and transferring passengers join the waiting queueon the platform and board a train based on a first-come-first-board (FIFB) discipline. The number ofsuccessfully boarding passengers depends on the available train capacity.3 a) Train arrival(b) Train departureFigure 2: Main functions of the event-based transit network loading model
The TNL model works by generating a train event list (arrivals and departures) based on the actual trainmovement data (AVL), and then sequentially processing the ordered events until all events are processed forthe time period of interest. The processing of an individual event is based on the following rules: • If the event is an arrival (Figure 2a), the train offloads passengers and updates its state (e.g., trainload and in-vehicle passengers). Alighting passengers who need to transfer are assigned to the waitingqueues on the corresponding transfer platforms (e.g., passengers transferring to platform B in Figure2a). Passengers who tap out will be removed from the system. New tap-in passengers who enteredthe station between two events are added into the queue (e.g., new tap-in passengers in platform A inFigure 2a). Then, the waiting queue objects for all platforms are updated accordingly. • If the event is a departure (Figure 2b), passengers board trains based on a FIFB priority rule. Ifthe on-board passengers reached the train capacity, the remaining passengers at the platform will bedenied boarding and wait for the next available train. Finally, the state of the train (train load andin-vehicle passengers) and the waiting queue at the platform are updated accordingly.More specifically, for each passenger in the simulation model, we first calculate his/her probability ofchoosing each available path based on the path’s attributes and path choice parameters (see Section 2.2 fordetails). Path attributes include in-vehicle time, number of transfers, transfer walking time, etc. Then eachpassenger is assigned with a specific path based on the choice probability. Based on the path information,4he passenger walks to a specific platform, joins the waiting queue and waits for available trains to board.The boarding and alighting behavior are as described above.
Consider a general urban rail network in a specific time period T , represented as G = ( S, A ), where S is the set of stations and A is the set of directed links. We divide T into several time intervals with equallength τ (e.g., τ = 15 min). Denote the set of all time intervals as T = { , , ..., T /τ } . Define a time-space(TS) node as i m , where i ∈ S and m ∈ T . i m represents station i in time interval m .For an OD pair ( i, j ) ( i, j ∈ S ), the OD entry flow ( q i m ,j ) represents the number of passengers enteringstation i during time interval m and exiting at station j . Let the set of all OD entry flows be q e . The ODexit flow ( q i,j n ) represents the number of passengers who exit at station j in the time interval n with origin i . q i m ,j and q i,j n are inputs and outputs of the TNL model, respectively.Let the set of all paths between ( i, j ) be R ( i, j ). We assume that the path choice behavior can beformulated as a C-logit model (Cascetta et al., 1996), which is an extension of the multinomial logit (MNL)model to correct the correlation among paths due to overlapping (Prato, 2009). The path choice fraction forpath r ∈ R ( i, j ) in time interval m ( p i m ,jr ) is formulated as: p i m ,jr = e µ ( β X · X r,m + β CF · CF r ) (cid:80) r (cid:48) ∈R ( i,j ) e µ ( β X · X r (cid:48) ,m + β CF · CF r (cid:48) ) , ∀ r ∈ R ( i, j ) , m ∈ T , i, j ∈ S (1)where µ is the scale parameter of the Gumbel distribution of the error term (Ben-Akiva et al., 1985), which isusually normalized to 1. Larger (smaller) µ means the choice behavior is more deterministic (random). X r,m is the vector of attributes for path r in time interval m (e.g., in-vehicle time, number of transfers, transferwalking time, etc.). CF r is the commonality factor of path r which measures the degree of similarity of path r with the other paths of the same OD. β X and β CF are the corresponding coefficients to be estimated. Let β be the vector that combines β X and β CF (i.e., β = [ β X , β CF ]). CF r is defined as: CF r = ln (cid:88) r (cid:48) ∈R ( i,j ) ( D r,r (cid:48) D r D r (cid:48) ) γ , (2)where D r,r (cid:48) is the number of common stations of path r and r (cid:48) . D r and D r (cid:48) are the number of stations forpath r and r (cid:48) , respectively. γ is a fixed positive parameter. Let the set of all path choice fractions be p .The values of β can be bounded from above and below. The boundaries can be obtained from the priorknowledge and previous survey results. Denote the upper bound as U β and lower bound as L β ( L β ≤ β ≤ U β ),where U β and L β are both vectors with the same cardinality as β .According to Mo et al. (2020c), the actual train capacity utilized by passengers is determined by threefactors: a) waiting passenger distribution on the platform, b) train load and distribution across the train,and c) passengers’ willingness to board a crowded train. Thus, train capacity is not constant. Instead, itis dynamic and changes across stations and trains depending on the crowding level of the train and theplatform. Mo et al. (2020c) model the capacity of train k at station i ( C k,i ) as: C k,i = (cid:40) θ n i + θ H k,i + θ Q k,i if station i is in the list of congested stations θ n i otherwise ∀ k, i (3)where n i is the number of cars of train i . H k,i is the load of train k when it arrives at station i . Q k,i is the number of passengers waiting on the platform when train k arrives at station i . θ , θ , and θ areparameters to be estimated ( θ , θ , θ > θ is a measure of service standard. θ n i can be seenas the base capacity, that is, the train load that represents acceptable service standards. At uncongestedstations, passengers are assumed not to board when the train load is greater than θ n i . At congested stations,passengers may still board a train even if it is already crowded (Liu et al., 2016), which makes the effectivetrain capacity higher than θ n i . θ captures the effect that the effective capacity is higher when train loadis higher. This is because passengers may worry if they did not board this crowded train, they cannot board5he following trains as well (Liu et al., 2016). θ captures the effect that more waiting passengers at theplatform may push more passengers to board, leading to higher effective capacity.In the discussion that follows, let θ be the vector of these three parameters. We assume that the valuesthat these parameters can take is L θ ≤ θ ≤ U θ , where L θ and U θ are the corresponding lower and upperbounds, respectively.The goal is to calibrate θ and β vectors (used by the TNL model) based on indirect observations. Two setsof observations are used for the calibration: observed OD exit flows and observed journey time distribution(JTD). Both of them can be obtained from the AFC data.Let the ground truth (observed) OD exit flow be ˜ q i,j n . Let f i,j t ( x ) be the model-derived JTD of passengerswith origin i who exit at station j during time interval t . Let ˜ f i,j t ( x ) be the corresponding observedJTD extracted from the AFC data. Since f i,j t ( x ) and ˜ f i,j t ( x ) are estimated from passengers’ journey timeobservations, only the OD pairs with more than E passengers exiting in a specific time interval are considered,where E is a predetermined threshold to ensure enough sample size. Denote the set of corresponding ODpairs and exit time intervals as E , where E = { ( i, j n ) : ˜ q i,j n , q i,j n > E, ∀ i, j ∈ S, n ∈ T } .The calibration problem is formulated as an optimization problem:min β, θ w (cid:88) i,j ∈ S, m ∈T ( q i,j n − ˜ q i,j n ) + w (cid:88) ( i,j n ) ∈E D KL ( f i,j n || ˜ f i,j n ) (4a)s . t . q i,j n = TNL( p , q e , θ ) ∀ i, j ∈ S, m ∈ T , (4b) f i,j n ( x ) = TNL( p , q e , θ ) ∀ ( i, j n ) ∈ E , (4c) p i m ,jr = e µ ( β X · X r,m + β CF · CF r ) (cid:80) r (cid:48) ∈R ( i,j ) e µ ( β X · X r (cid:48) ,m + β CF · CF r (cid:48) ) ∀ p i m ,jr ∈ p , (4d) L β ≤ β ≤ U β , (4e) L θ ≤ θ ≤ U θ (4f)The objective function (Eq. 4a) has two parts: the square error between model-derived OD exit flows andthe corresponding observations, and the difference between model-derived and observed JTD. w and w areweights used to balance the scale and the importance of the two parts. The difference of the two distributionsis expressed using Kullback-Leibler (KL) divergence ( D KL ): D KL ( f i,j n || ˜ f i,j n ) = (cid:90) x f i,j n ( x ) · log f i,j n ( x )˜ f i,j n ( x ) d x. (5)TNL( p , q e , θ ) is the black-box function that corresponds to the TNL model, which can output the model-derived OD exit flows and JTD for a given set of path choices and train capacity. Since the TNL model hasno analytic form, Eq. 4 is a SBO problem with upper and lower bound constraints. In the following section,we discuss seven different algorithms appropriate for the solution of SBO problems. These algorithms belongto four general approaches of SBO solving methods.It is worth noting that X r,m (i.e., the path attributes vector) is known and fixed in this study. It isassumed to represent the historical path conditions based on which passengers make their habitual choices.Different from typical transit/traffic assignment problems where path choices are estimated by assuming userequilibrium (for planning purposes), the AFC data-based estimation aims to find the actual realized pathchoices based on real-world observations (i.e., OD entry-exit flows). Since passengers make decisions beforeknowing the actual travel or waiting times, X r,m should reflect passengers’ historical perceptions of pathattributes and should not change within the model estimation process. Therefore, though C k,i captures theactual path crowding information, it should not be included in the path choice formulation as passengersmake decisions before knowing the actual crowding.
3. Simulation-based optimization algorithms
There are four major classes of methods for solving the SBO problems, including the heuristic methods,direct search methods, gradient-based methods, and response surface methods (Osorio and Bierlaire, 2013;6maran et al., 2016). Heuristic methods are partial search algorithms that may provide a sufficientlygood solution to an optimization problem, especially with incomplete or imperfect information or limitedcomputation capacity. Direct search methods are derivative-free methods that are based on the sequentialexamination of trial points generated by a certain strategy. They are attractive as they are easy to describeand implement. More importantly, they are suitable for objective functions where gradients do not existeverywhere. Gradient-based approaches (or stochastic approximation methods) attempt to optimize theobjective function using estimated gradient information. These methods aim to imitate the steepest descentmethods in derivative-based optimization. Finite difference schemes can be used to estimate gradientsbut they may involve a large number of expensive function evaluations if the number of decision variables islarge. Response surface methods are useful in the context of continuous optimization problems. They focus onlearning input-output relationships to approximate the underlying simulation by a predefined functional form(also known as a meta-model or surrogate model). This functional form can then be used for optimizationleveraging powerful derivative-based optimization techniques.In this study, we use seven representative algorithms belonging to these four classes of SBO methodsto address the aforementioned path choice and train capacity calibration problem. Table 1 summarizes themain characteristic of these algorithms. The summary of all algorithms is described in Table 1.In the discussion that follows, let Θ be the combined vector of β and θ (i.e., Θ = [ β, θ ] is the vector ofall coefficients to be estimated). Let N be the dimension of Θ (i.e., Θ ∈ R N ). Table 1: Algorithms Summary
Type Algorithm Constraints Stochastic SourceHeuristic method Genetic Algorithm (GA) Yes Yes Fortin et al. (2012)Simulated Annealing (SA) Yes Yes Tsallis and Stariolo (1996)Direct search Nelder-Mead Simplex Algorithm (NMSA) No No Gao and Han (2012)Mesh Adaptive Direct Search (MADS) Yes Yes Abramson et al. (2009)Gradient-based Simultaneous PerturbationStochastic Approximation (SPSA) Yes Yes Spall et al. (1992)Response surface Bayesian Optimization (BYO) Yes Yes Snoek et al. (2012)Constrained Optimization usingResponse Surfaces (CORS) Yes Yes Regis and Shoemaker (2005)
GA is a heuristic method for solving both constrained and unconstrained optimization problems, whichbelongs to the larger class of evolutionary algorithms inspired by natural selection, the process that drivesbiological evolution. The GA repeatedly modifies a population of individual solutions as an evolution process(Whitley, 1994). The GA can be used to solve a variety of optimization problems that are not well suited forstandard optimization algorithms, such as the SBO problem where the objective function (or constraints) isnondifferentiable and highly nonlinear.The evolution starts from a population of randomly generated individuals, and is an iterative process,with the population in each iteration called a generation. In each generation, the genetic algorithm selectsindividuals at random from the current population to be parents and uses them to produce the children forthe next generation. Over successive generations, the population ”evolves” toward an optimal solution. Thegenetic algorithm uses three main procedures at each step to create the next generation from the currentpopulation: 1) Selection: select the individuals, called parents, that contribute to the population of the nextgeneration. Individuals with better objective function values are more likely to be selected. 2) Crossover:combine two parents to form children for the next generation. 3) Mutation: apply random changes toindividual parents to form children.In this study, we adopted a blend crossover and Gaussian mutation methods. The probability of crossoveris set as 0.8 and the probability of mutating is set as 0.4. And the population size is set as 6 given the limitedcomputational budget. The algorithm is implemented by the Python deap package (Fortin et al., 2012).7 .2. Simulated Annealing (SA)
SA is a heuristic method for solving optimization problems (Van Laarhoven and Aarts, 1987). Themethod is based on the physical process of heating a material and then slowly lowering the temperature todecrease defects, thus minimizing the system energy.At each iteration of the SA algorithm, a new point is randomly generated. The distance of the newpoint from the current point, or the extent of the search, is based on a probability distribution with a scaleproportional to the temperature. A distorted Cauchy-Lorentz visiting distribution is used in this study(Tsallis and Stariolo, 1996). The algorithm accepts all new points that lower the objective function, butalso, with a certain probability, points that raise the objective function. By accepting points that raise theobjective function, the algorithm avoids being trapped in local minima. An annealing schedule is selectedto systematically decrease the temperature as the algorithm proceeds. As the temperature decreases, thealgorithm reduces the extent of its search to converge to a minimum.In this study, the SA algorithm in Python
Scipy package is adopted for the implementation with allmodel parameters set as default (Scipy, 2019).
NMSA is a simplex method for finding a local minimum (Nelder and Mead, 1965). NMSA in N dimensionsmaintains a set of N + 1 test points arranged as a simplex . Denote the initial value of Θ as Θ ini . The initialsimplex set ( N + 1 points) is generated as { Θ : Θ = Θ ini + e i , ∀ i = 1 , ..., N } ∪ { Θ ini } , where e i ∈ R N is theunit vector in the i th coordinate, σ is the step-size which is set as 0.05 in this study (Gao and Han, 2012).Based on the initial simplex, the model evaluates the objective function for each test point, in order tofind a new test point to replace one of the old test points. The new candidate can be generated throughsimplex centroid reflections, contractions, or other means depending on the function value of the test points.The process will generate a sequence of simplexes, for which the function values at the vertices get smallerand smaller. The size of the simplex is reduced and finally, the coordinates of the minimum point are found.Four possible operations: reflection, expansion, contraction, and shrink are associated with the corre-sponding scalar parameters: α (reflection), α (expansion), α (contraction) and α (shrink). In this study,we set the value of these parameters as { α , α , α , α } = { , , . , . } as suggested in Gao and Han (2012).The algorithm is implemented by the Python scikit-learn package with all parameters set as default. SinceNMSA is designed for unconstrained problem, we turned the bound of Θ into a big penalized term in theobjective function for this algorithm. More details regarding the NMSA can be referred to (Gao and Han,2012). The MADS algorithm is a direct search framework for nonlinear optimization (Audet and Dennis Jr,2006). It seeks to improve the current solution by testing points in the neighborhood of the current point(the incumbent). The neighborhood points are generated by moving one step in each direction from theincumbent on an iteration-dependent mesh. Each iteration of MADS consists of a SEARCH stage and anoptional POLL stage. The SEARCH stage evaluates a finite number of points proposed by the searchingstrategy (e.g. moving one step around from the current point). Whenever the SEARCH step fails to generatean improved mesh point, the POLL step is invoked. The POLL step conducts local exploration near thecurrent incumbent, which also intends to find an improved point on the mesh. Once an improved pointis found, the algorithm updates the current point and constructs a new mesh. According to (Audet andDennis Jr, 2006), the mesh size parameters approach zero as the number of iteration approaches infinity,which demonstrates the convergence of the MADS algorithm.In this paper, we use a variant of the MADS method called ORTHO-MADS, which leverages a specialorthogonal positive spanning set of polling directions. More details regarding the algorithm can be foundin (Abramson et al., 2009). NOMAD 3.9.1 (Audet et al., 2009) with the Python interface is used for theMADS algorithm application. The hyper-parameters are tuned based on the NOMAD user guide. Thedirection type is set as orthogonal, with N + 1 directions generated at each poll. Latin Hypercube search isnot applied. 8 .5. Simultaneous Perturbation Stochastic Approximation (SPSA) SPSA is a descent direction method for finding local minimums. It approximates the gradient with onlytwo measurements of the objective function, regardless of the dimension of the optimization problem. Denotethe objective function in Eq. 4 as Z (Θ). The estimated parameters in the k -th iteration is denoted as Θ ( k ) .Then one iteration for the SPSA is performed asΘ ( k +1) = Θ ( k ) − a k · ˆ ∇ Z (Θ ( k ) ) (6)where ˜ ∇ Z (Θ ( k ) ) = Z (Θ ( k ) + c k ∆ k ) − Z (Θ ( k ) − c k ∆ k )2 c k ∆ k (7) a k = a ( k + 1 + A ) α (8) c k = c ( k + 1) γ (9)∆ k is a random perturbation vector, whose elements are obtained from a Bernoulli distribution with theprobability parameter equal to 0.5. { α, γ, a, c, A } are tuned as { . , . , . , . , . M } in this studyaccording to the numerical tests and guidelines from prior empirical studies (Gomez-Dans, 2012). M is themaximum number of iterations. BYO constructs a probabilistic model of the objective function and exploits this model to determinewhere to evaluate the objective function for the next step. The philosophy of BYO is to use all of theinformation available from previous evaluations, instead of simply relying on the local gradient and Hessianapproximations. This enables BYO to find the minimum of difficult non-convex functions with relativelyfew function evaluations.BYO assumes a prior distribution for the objective function values and uses an acquisition function todetermine the next point to evaluate. In this study, we use the Gaussian process as the prior distributionfor the objective function due to its flexibility and tractability. For the acquisition function, we tested threecommon criteria: probability of improvement (POI), expected improvement (EI), and upper confidencebound (UCB) (Snoek et al., 2012). The EI criterion is used in this path choice calibration problem due to itsbest performance in our problem. The BYO is implemented in Python with the bayes opt package. Moredetails regarding the BYO can be found in (Snoek et al., 2012).
CORS is a response surface method for global optimization. In each iteration, it updates the responsesurface model based on all previously probed points and selects the next point to evaluate. The principles fornext point selection are (a) finding new points that have lower objective function value, and (b) improvingthe fitting of the response surface model by sampling feasible regions where little information exists. Hence,the next point is selected by solving the minimization problem of the current response surface functionsubject to constraints that the next point should be more than a certain distance away from all previouspoints (Regis and Shoemaker, 2005).An algorithm following the CORS framework requires two components: (a) a scheme for selecting aninitial set of points for objective function evaluation and (b) a procedure for globally approximating theobjective function (i.e., a response surface model). In this study, the initial sampling is conducted using theLatin hypercube methods, with the initial sampling number equal to 0.2 × the total number of functionevaluations allowed. The radial basis function (RBS) is used as the response surface model. For the subse-quent sampling, a modified version of the CORS algorithm with space re-scaling is used. Details about thealgorithm can be found in Regis and Shoemaker (2005) and Knysh and Korkolis (2016).9 . Case study The proposed modeling framework is tested using data from the Hong Kong MTR network. MTR is theoperator of the Hong Kong urban rail network, which provides services for the urbanized areas of Hong KongIsland, Kowloon, and the New Territories. The system currently consists of 11 lines with 218.2 km (135.6miles) of rail, serving 159 stations including 91 heavy rail stations and 68 light rail stops. It serves over 5million trips on an average weekday. Most of the passengers use a smart card fare-payment system namedOctopus. For the urban heavy rail lines, trip transactions are recorded when passengers enter and exit thesystem, providing information about the tap-in and tap-out stations and corresponding timestamps.
We use AFC data on a typical weekday afternoon peak period (18:00-19:00) in March 2017 for themodel application. Li (2014) conducted a revealed-preference (RP) path choice survey of more than 20,000passengers in the MTR system and used them to estimate a path choice model. The estimation resultsare shown in Appendix A. The following attributes were used in the specification of the model: (a) totalin-vehicle time, (b) the number of transfer times, (c) relative walking time (total walking time divided bytotal path distance), and (d) the commonality factor (Eq. 2). Future research may consider more pathchoice attributes such as perceived crowding levels, estimated waiting times, etc.As the real-world path choice information and train capacity are usually unavailable, we validate themodels with synthetic data. To generate the synthetic data, we first extract the OD entry flow ( q i m ,j )from the real-world AFC records. We assume a synthetic Θ as the “true” path choice and train capacityparameters. The TNL model with the true OD entry flow, train timetable, and the synthetic Θ as inputsis used to simulate the travel of passengers in the system and record people’s tap-in and tap-out time. Theinput timetable is treated as the synthetic AVL data . The resulting passengers’ tap-in and tap-out times aretreated as the synthetic AFC data . The synthetic data, including “true” passenger path choices and traincapacity, are used to evaluate the performance of the model under the various solution algorithms. All ODpairs of the whole network are considered in the experiments.To compare the different SBO solving algorithms, we design five test scenarios summarized in Table 2.Each scenario has a different synthetic Θ. The selection of synthetic Θ can represent different assumptionsabout passengers’ choice behavior and sensitivity to crowding. For the reference scenario, we use the pathchoice parameters in Table A.8 as the synthetic β and use the estimated train capacity parameters in Moet al. (2020c) as the synthetic θ . Table 2: Scenario design
Parameter category Synthetic Θ Scenarios BoundReference Path choice Train capacityRandom Deterministic Crowding-sensitive Crowding-insensitivePath choice In vehicle time -0.147 0 -2.0 -0.147 -0.147 [-2, 0]Relative walking time -1.271 0 -5.0 -1.271 -1.271 [-5, 0]Number of transfers -0.573 0 -3.0 -0.573 -0.573 [-3, 0]Commonality factor -3.679 0 -10.0 -3.679 -3.679 [-10, 0]Train capacity θ
232 232 232 225 235 [220, 260] θ θ Passengers’ actual path choice behavior is assumed to be random (each path is equally likely to beselected) or deterministic. For the random path choice scenario, we set all synthetic choice parameters as 0,which means all available paths are equally likely to be chosen. For the deterministic path choice scenario, The word ”deterministic” here just represents the degree of randomness is low. The “truly” deterministic corresponds to allparameters go to → −∞
10e set all synthetic choice parameters as the lower bounds (i.e., the maximum absolute value possible).Under this scenario, a slight difference in attributes between two paths can lead to a high difference inchoice probability (i.e., this is close to passengers following the shortest path). As for the train capacity, thesynthetic θ for these two scenarios is the same as the reference scenario.Passengers’ sensitivity to crowding may also vary. If all passengers are not sensitive to the crowding, traincapacity can be modeled as a fixed value. However, if passengers become more sensitive to the crowding, theactual train capacity may largely depend on the crowding level in the train and on the platform. Therefore,passengers’ sensitivity to crowding can be reflected by the scale of θ and θ (Mo et al., 2020c). For thecrowding-sensitive scenario, we set the synthetic train capacity parameters as θ = 225 , θ = 0 . , θ = 0 . θ and θ are higher to represent higher sensitivity. And θ is decreasedto offset the capacity increase caused by the increase of θ and θ . As for the crowding-insensitive scenario, weset the synthetic train capacity parameters as θ = 235 , θ = 0 , θ = 0, which can be seen as a fixed-capacitymodel. The lower and upper bounds of all parameters ( L β , U β , L θ , U θ ) are shown in Table 2. Θ ini is set as( L Θ + U Θ ) / The convergence results of the reference scenario are depicted in Figure 3. Each point represents theaverage value over all replications. We found that the performance of different algorithms varied. Giventhe limited number of function evaluations, CORS, BYO, and SPSA converge to a relatively small objectivefunction. GA, MADS, and SA have relatively large objective function values upon termination. In terms ofconvergence speed, the response surface methods (BYO and CORS) have the fastest convergence speed. Theyalso reach the lowest objective function value. This is consistent with conclusions regarding the performanceof the SBO algorithms when used in the transportation domains (Osorio and Bierlaire, 2013; Cheng et al.,2019; Mo et al., 2020b; Mo, 2020).Figure 3 also summarizes the behavior of the algorithm stability. The vertical line indicates the × standard deviations over the five replications. NMSA is a deterministic algorithm and not affected byrandomness. BYO and CORS show high randomness in the first half iterations. However, as the numberof function evaluations increases, the standard deviation of the objective function decreases, and the resultsbecome stable. GA, SA, and MADS are unstable compared to other algorithms. This means that theheuristic algorithms (GA and SA) are not suitable for the calibration problem studied in this paper. Theinstability of MADS may be because it may converge to non-stationary points (Abramson and Audet, 2006).Table 3 compares the parameters estimated by different algorithms with the synthetic ones. Althoughsome algorithms can reach similar objective function values, they result in different estimated parameters.For example, CORS and SPSA have similar objective function values. However, SPSA performs better inpath choice estimation while CORS performs better in train capacity estimation. We also observe that thetrain capacity parameters are relatively harder to estimate. This may be because most of the stations in therail system are not congested and all passengers can board the trains. Thus, the objective function is notvery sensitive to train capacity parameters. 11 igure 3: Convergence results of reference scenario. The error bar indicates × standard deviation. NMSA has no error barbecause it is a deterministic algorithm.Table 3: estimation results of the reference scenario Category Variable name ”True” Estimated parametersGA SA NMSA MADS SPSA BYO CORSPath choice In-vehicle time -0.147 -0.392 -0.327 -0.342 -0.454 -0.170 -0.207 -0.229Relative walking time -1.271 -2.205 -3.010 -3.020 -0.302 -2.257 -2.493 -2.486Number of transfers -0.573 -1.143 -0.787 -0.389 -1.248 -0.598 -0.776 -0.756Commonality factor -3.679 -6.482 -6.851 -7.250 -7.834 -4.419 -5.434 -5.716Train capacity θ
232 239 243 259 252 241 234 243 θ θ Figure 4 shows the estimation results for the two path choice related scenarios: random and deterministic.The estimated parameters are shown in Table 4 and 5. For the random scenario, all “true” (synthetic) pathchoice parameters are set as zero, which means all paths are equally likely to be chosen. We observe that,in this scenario (Figure 4a), CORS and SA algorithms perform the best with the lowest objective function.Compared to the reference scenario in Section 4.3, the decreased performance of BYO and SPSA may bedue to the ”true” β is close to the upper-bound ( U β = 0). The Gaussian posterior distribution in BYOand gradient estimation in SPSA can suffer from instability in the boundary. From Table 4, we observe theparameters of in-vehicle time and number of transfers are better estimated than those of relative walkingtime and commonality factors.Figure 4b shows the results of the deterministic scenario. The initial objective function is relativelysmall (1 . × ) compared to the reference scenario (1 . × ). All algorithms only reduce the objectivefunction by around except for the CORS algorithm. The good performance of CORS may come fromglobal searching with the Latin hypercube method. It is better suited to explore the points near boundaries.Although the objective function does not decrease too much, the estimated parameters are still acceptable(see Table 5). 12 able 4: estimation results of the random path choice scenario Category Variable Name ”True” Estimated ParametersGA SA NMSA MADS SPSA BYO CORSPath choice In-vehicle time 0 0 -0.072 -0.050 0 -0.108 -0.037 0Relative walking time 0 -2.151 -1.139 -1.807 -1.000 -1.719 -3.725 -0.702Number of transfers 0 -0.348 -0.185 -0.435 -1.334 -0.631 -0.207 0Commonality factor 0 -5.997 -1.945 -9.991 -5.432 -5.127 -4.155 -8.000Train capacity θ
232 243 224 254 232 241 248 223 θ θ Table 5: estimation results of the deterministic path choice scenario
Category Variable Name ”True” Estimated ParametersGA SA NMSA MADS SPSA BYO CORSPath choice In-vehicle time -2 -1.240 -1.243 -1.205 -1.160 -1.544 -1.537 -1.830Relative walking time -5 -3.180 -3.358 -2.819 -2.480 -3.728 -3.807 -4.492Number of transfers -3 -1.575 -1.551 -1.419 -1.524 -1.786 -1.761 -2.661Commonality factor -10 -5.307 -5.251 -4.735 -4.920 -6.346 -6.379 -8.819Train capacity θ
232 237 232 228 237 239 232 237 θ θ a) Random(b) DeterministicFigure 4: Algorithm performance in the two path choice scenarios Figure 5 shows the estimation results of the two scenarios related to train capacity (i.e., crowding-sensitiveand crowding-insensitive). In the crowding-insensitive scenario (Figure 5a), the conclusions are similar tothe reference scenario. CORS, BYO, NMSA, and SPSA converge to low objective function values andoutperform other algorithms. The performance of NMSA and MADS is improved compared to the referencescenario. In the crowding-sensitive scenario, we still observe a good performance by the CORS, NMSA, andSPSA algorithms. The performance of BYO is slightly reduced. The results shown in Table 6 and 7 indicatethat θ (base capacity) is hard to estimate. This may be because trains at most stations do not reach thecapacity. Therefore, for many OD pairs, the OD exit flows (directly related to the objective function) arenot sensitive to the base capacity parameter. 14 a) Crowding-insensitive(b) Crowding-sensitiveFigure 5: Algorithm performance in the two train capacity scenariosTable 6: estimation results of the crowding-insensitive train capacity scenario Category Variable Name ”True” Estimated ParametersGA SA NMSA MADS SPSA BYO CORSPath choice In-vehicle time -0.147 -0.392 -0.181 -0.254 -0.460 -0.191 -0.197 -0.177Relative walking time -1.271 -2.153 -2.044 -2.636 -2.294 -2.284 -2.469 -2.025Number of transfers -0.573 -1.127 -1.614 -1.279 -0.490 -0.760 -0.908 -1.011Commonality factor -3.679 -6.489 -6.500 -7.492 -7.750 -5.299 -5.474 -5.130Train capacity θ
235 239 245 249 230 241 238 236 θ θ able 7: estimation results of the crowding-sensitive train capacity scenario Category Variable Name ”True” Estimated ParametersGA SA NMSA MADS SPSA BYO CORSPath choice In-vehicle time -0.147 -0.472 -0.217 -0.228 -0.332 -0.177 -0.195 -0.196Relative walking time -1.271 -2.533 -1.575 -2.735 -1.568 -2.118 -1.763 -2.534Number of transfers -0.573 -0.759 -1.169 -1.323 -0.816 -0.495 -0.892 -0.734Commonality factor -3.679 -6.489 -6.324 -7.040 -7.834 -4.361 -6.046 -5.021Train capacity θ
225 238 244 238 245 244 237 237 θ θ
5. Conclusion
In this paper, we propose an SBO framework to calibrate train capacity and path choice model parameterssimultaneously for metro systems using AFC and AVL data. The advantage of the proposed framework liesin capturing the collective effect of both path choices and train capacity on passenger journey times. Sevenrepresentative algorithms from four main branches of SBO methods are applied and compared with respectto their solution accuracy, convergence speed, and stability. We applied the proposed framework using datafrom the Hong Kong MTR network and compared the performance of the different algorithms. Overall, theresults show that some algorithms result in a reasonable estimation of the parameters of interest. Theseresults also support the effectiveness of the proposed SBO framework for calibrating these key parametersusing AFC and AVL data. Especially, the response surface methods (particularly CORS) exhibit consistentlygood performance. The SBO framework is flexible to accommodate a wide range of path choice and traincapacity models in transit simulation models.This paper has some limitations. First, we validate the framework and evaluate the algorithmic perfor-mance only using synthetic AFC and AVL data. Therefore, the complexities of noise and uncertainties inactual data do not play any role. This is caused by the absence of real-world path choice and train capacityinformation. Future research can collect real-world path choice and train capacity data to conduct morerealistic model validation. Second, we assumed that the path choice behavior is similar for the whole net-work (same β values). Given the real-world path choice behavior is possibly more diverse and heterogeneous,future research can explore clustering different OD pairs with different β values based on individual mobilitycharacteristics (Mo et al., 2021).
6. Author contribution statement
The authors confirm contribution to the paper as follows: study conception and design: B. Mo, Z. Ma,H.N. Koutsopoulos, J. Zhao; data collection: B. Mo, Z. Ma; analysis and interpretation of results: B. Mo,Z. Ma, H.N. Koutsopoulos; draft manuscript preparation: B. Mo, H.N. Koutsopoulos. All authors reviewedthe results and approved the final version of the manuscript.
7. Acknowledgements
The authors would like to thank the Hong Kong Mass Transit Railway (MTR) for their support and dataavailability for this research. Also, we acknowledge MIT Libraries for providing funding for the open-accesspublication of the paper. The early presentation (preprint) of the manuscript is in ResearchGate.
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Appendices
Appendix A. Passenger path Choice Model for MTR System
These results are from (Li, 2014). The C-logit Model formulation is the same as Eq. (1) and Eq. (2). Atotal number of 31,640 passengers completed the questionnaire. After filtering duplicate responses, 26,996responses were available. The model results are shown in Table A.8. The main explanatory variables arethe total in-vehicle time, relative transfer walking time, and the number of transfers. All variables arestatistically significant with the expected signs. Paths with high in-vehicle time, walking time, and thenumber of transfers are less likely to be chosen by passengers.
Table A.8: Path Choice Model Estimation Results
Estimate Std. Error t-valueIn-vehicle time -0.147 0.011 -13.64 ***Relative walking time -1.271 0.278 -4.56 ***Number of transfers -0.573 0.084 -6.18 ***Commonality factor -3.679 1.273 -2.89 ** ρ = 0 . p < .
01; **: p < ..