User perspectives in public transport timetable optimisation

The present paper deals with timetable optimisation from the perspective of minimising the waiting time experienced by passengers when transferring either to or from a bus. Due to its inherent complexity, this bi-level minimisation problem is extremely difficult to solve mathematically, since timetable optimisation is a non-linear non-convex mixed integer problem, with passenger flows defined by the route choice model, whereas the route choice model is a non-linear non-continuous mapping of the timetable. Therefore, a heuristic solution approach is developed in this paper, based on the idea of varying and optimising the offset of the bus lines. Varying the offset for a bus line impacts the waiting time passengers experience at any transfer stop on the bus line.In the bi-level timetable optimisation problem, the lower level is a transit assignment calculation yielding passengers’ route choice. This is used as weight when minimising waiting time by applying a Tabu Search algorithm to adapt the offset values for bus lines. The updated timetable then serves as input in the following transit assignment calculation. The process continues until convergence.The heuristic solution approach was applied on the large-scale public transport network in Denmark. The timetable optimisation approach yielded a yearly reduction in weighted waiting time equivalent to approximately 45 million Danish kroner (9 million USD).     

A bi-level model for single-line rail timetable design with consideration of demand and capacity

This paper proposes a bi-level model to solve the timetable design problem for an urban rail line. The upper level model aims at determining the headways between trains to minimize total passenger cost, which includes not only the usual perceived travel time cost, but also penalties during travel. With the headways given by the upper level model, passengers’ arrival times at their origin stops are determined by the lower level model, in which the cost-minimizing behavior of each passenger is taken into account. To make the model more realistic, explicit capacity constraints of individual trains are considered. With these constraints, passengers cannot board a full train, but wait in queues for the next coming train. A two-stage genetic algorithm incorporating the method of successive averages is introduced to solve the bi-level model. Two hypothetical examples and a real world case are employed to evaluate the effectiveness of the proposed bi-level model and algorithm. Results show that the bi-level model performs well in reducing total passenger cost, especially in reducing waiting time cost and penalties. And the section loading-rates of trains in the optimized timetable are more balanced than the even-headway timetable. The sensitivity analyses show that passenger’s desired arrival time interval at destination and crowding penalty factor have a high influence on the optimal solution. And with the dispersing of passengers' desired arrival time intervals or the increase of crowding penalty factor, the section loading-rates of trains become more balanced.     

A stochastic optimization model for transit network timetable design to mitigate the randomness of traveling time by adding slack time

Transit network timetabling aims at determining the departure time of each trip of all lines in order to facilitate passengers transferring either to or from a bus. In this paper, we consider a bus timetabling problem with stochastic travel times (BTP-STT). Slack time is added into timetable to mitigate the randomness in bus travel times. We then develop a stochastic integer programming model for the BTP-STT to minimize the total waiting time cost for three types of passengers (i.e., transferring passengers, boarding passengers and through passengers). The mathematical properties of the model are characterized. Due to its computational complexity, a genetic algorithm with local search (GALS) is designed to solve our proposed model (OPM). The numerical results based on a small bus network show that the timetable obtained from OPM reduces the total waiting time cost by an average of 9.5%, when it is tested in different scenarios. OPM is relatively effective if the ratio of the number of through passengers to the number of transferring passengers is not larger than a threshold (e.g., 10 in our case). In addition, we test different scale instances randomly generated in a practical setting to further verify the effectiveness of OPM and GALS. We also find that adding slack time into timetable greatly benefits transferring passengers by reducing the rate of transferring failure.     

Methods for creating bus timetables

This paper provides alternative methods for constructing bus timetables using passenger load data. It attempts to fulfill six major objectives: to evaluate alternative timetables in terms of required resources; to improve the correspondence of bus departure times with passenger demand; to provide alternative timetables for the schedulers' use in specific scheduling situations; to permit direct bus frequency changes for possible exceptions (known to the schedulers) which do not rely on passenger demand data; to allow the construction of timetables with headway smoothing techniques (similar to that performed manually); and to integrate different headway setting and different timetable construction methods. The procedures developed set the bus departure times for the case of evenly spaced headways and for the case of allowing the headways to be unevenly spaced. In the first case, smoothing techniques are developed in the transition segments between adjacent time periods. In the second case, the departure times are shifted so as to obtain uniform average loads instead of even headways. The final product of the research consists of a set of computer programs which are tested on a heavy bus line in Los Angeles.     

A self-adaptive method to equalize headways: Numerical analysis and comparison

In uncontrolled bus systems, buses tend to bunch due to the stochastic nature of traffic flows and passenger demand at bus stops. Although schedules and priori target methods introduce slack time to delay buses at control points to maintain constant headways between successive buses, too much slack required delay passengers on-board. In addition, these methods focus on regular headways and do not consider the rates of convergence of headways after disturbances. We propose a self-adaptive control scheme to equalize the headways of buses with little slack in a single line automatically. The proposed method only requires the information from the current bus at the control point and both its leading and following buses. This elegant method is shown to regulate headways faster than existing methods. In addition, compared to previous self-equalizing methods, the proposed method can improve the travel time of buses by about 12%, while keeping the waiting time of passengers almost the same.     

An activity-based approach for scheduling multimodal transit services

This paper proposes a new activity-based transit assignment model for investigating the scheduling (or timetabling) problem of transit services in multi-modal transit networks. The proposed model can be used to generate the short-term and long-term timetables of multimodal transit lines for transit operations and service planning purposes. The interaction between transit timetables and passenger activity-travel scheduling behaviors is captured by the proposed model, as the activity and travel choices of transit passengers are considered explicitly in terms of departure time choice, activity/trip chain choices, activity duration choice, transit line and mode choices. A heuristic solution algorithm which combines the Hooke–Jeeves method and an iterative supply–demand equilibrium approach is developed to solve the proposed model. Two numerical examples are presented to illustrate the differences between the activity-based approach and the traditional trip-based method, together with comparison on the effects of optimal timetables with even and uneven headways. It is shown that the passenger travel scheduling pattern derived from the activity-based approach is significantly different from that obtained by the trip-based method, and that a demand-sensitive (with uneven headway) timetable is more efficient than an even-headway timetable.     

Equity-based timetable synchronization optimization in urban subway network

In the urban subway transportation system, passengers may have to make at least one transfer traveling from their origin to destination. This paper proposes a timetable synchronization optimization model to optimize passengers’ waiting time while limiting the waiting time equitably over all transfer station in an urban subway network. The model aims to improve the worst transfer by adjusting the departure time, running time, the dwelling time and the headways for all directions in the subway network. In order to facilitate solution, we develop a binary variables substitute method to deal with the binary variables. Genetic algorithm is applied to solve the problem for its practicality and generality. Finally, the suggested model is applied to Beijing urban subway network and several performance indicators are presented to verify the efficiency of suggested model. Results indicate that proposed timetable synchronization optimization model can be used to improve the network performance for transfer passengers significantly.     

Stochastic transit equilibrium

We present a transit equilibrium model in which boarding decisions are stochastic. The model incorporates congestion, reflected in higher waiting times at bus stops and increasing in-vehicle travel time. The stochastic behavior of passengers is introduced through a probability for passengers to choose boarding a specific bus of a certain service. The modeling approach generates a stochastic common-lines problem, in which every line has a chance to be chosen by each passenger. The formulation is a generalization of deterministic transit assignment models where passengers are assumed to travel according to shortest hyperpaths. We prove existence of equilibrium in the simplified case of parallel lines (stochastic common-lines problem) and provide a formulation for a more general network problem (stochastic transit equilibrium). The resulting waiting time and network load expressions are validated through simulation. An algorithm to solve the general stochastic transit equilibrium is proposed and applied to a sample network; the algorithm works well and generates consistent results when considering the stochastic nature of the decisions, which motivates the implementation of the methodology on a real-size network case as the next step of this research.     

Analysis of real-time control strategies in a corridor with multiple bus services

Control strategies have been widely used in the literature to counteract the effects of bus bunching in passenger‘s waiting times and its variability. These strategies have only been studied for the case of a single bus line in a corridor. However, in many real cases this assumption does not hold. Indeed, there are many transit corridors with multiple bus lines interacting, and this interaction affects the efficiency of the implemented control mechanism.This work develops an optimization model capable of executing a control scheme based on holding strategy for a corridor with multiple bus lines.We analyzed the benefits in the level of service of the public transport system when considering a central operator who wants to maximize the level of service for users of all the bus lines, versus scenarios where each bus line operates independently. A simulation was carried out considering two medium frequency bus lines that serve a set of stops and where these two bus lines coexist in a given subset of stops. In the simulation we compared the existence of a central operator, using the optimization model we developed, against the independent operation of each line.In the simulations the central operator showed a greater reduction in the overall waiting time of the passengers of 55% compared to a no control scenario. It also provided a balanced load of the buses along the corridor, and a lower variability of the bus headways in the subset of stops where the lines coexist, thus obtaining better reliability for all types of passengers present in the public transport system.     

Creating bus timetables with maximal synchronization

This paper addresses the problem of generating a timetable for a given network of buses so as to maximize their synchronization. It attempts to maximize the number of simultaneous bus arrivals at the connection (transfer) nodes of the network. Transit schedulers, taking into account the satisfaction and convenience of the system's users, appreciate the importance of creating a timetable with maximal synchronization, which enables the transfer of passengers from one route to another with minimum waiting time at the transfer nodes. In this paper, the problem is formulated as a mixed integer linear programming problem, and a heuristic algorithm is developed to solve the problem in polynomial time. The efficiency of this algorithm, compared to optimal solutions, is illustrated through a series of examples.     

Optimization of airport bus timetable in cultivation period considering passenger dynamic airport choice under conditions of uncertainty

An airport bus service, which is newly introduced in a multi-airport region, commonly leads to a gradually increasing market share of airports until a new state of equilibrium is reached. With the goal of speeding up and enlarging the increase in market share, this paper proposes a timetable optimization model by incorporating reactions of airport-loyal passengers to bus service quality. The simulation part of the model, which uses cumulative prospect theory to formulate discrete airport choices, results in predicted passenger demand needed in the optimization part. Then a genetic algorithm for multi-objective optimization problems called NSGA-II is applied to solve the model. To illustrate the model, the “Lukou airport-Wuxi” airport bus in China is taken as an example. The results show that the optimized timetables shorten the cultivation period and impel the market share to grow rapidly.     

Evaluation of an existing bus network using a transit network optimisation model: a case study of the Hiroshima City Bus network

This study evaluates an existing bus network from the perspectives of passengers, operators, and overall system efficiency using the output of a previously developed transportation network optimisation model. This model is formulated as a bi-level optimisation problem with a transit assignment model as the lower problem. The upper problem is also formulated as bi-level optimisation problem to minimise costs for both passengers and operators, making it possible to evaluate the effects of reducing operator cost against passenger cost. A case study based on demand data for Hiroshima City confirms that the current bus network is close to the Pareto front, if the total costs to both passengers and operators are adopted as objective functions. However, the sensitivity analysis with regard to the OD pattern fluctuation indicates that passenger and operator costs in the current network are not always close to the Pareto front. Finally, the results suggests that, regardless of OD pattern fluctuation, reducing operator costs will increase passenger cost and increase inequity in service levels among passengers.     

Demand-driven timetable design for metro services

Timetable design is crucial to the metro service reliability. A straightforward and commonly adopted strategy in daily operation is a peak/off-peak-based schedule. However, such a strategy may fail to meet dynamic temporal passenger demand, resulting in long passenger waiting time at platforms and over-crowding in trains. Thanks to the emergence of smart card-based automated fare collection systems, we can now better quantify spatial–temporal demand on a microscopic level. In this paper, we formulate three optimization models to design demand-sensitive timetables by demonstrating train operation using equivalent time (interval). The first model aims at making the timetable more dynamic; the second model is an extension allowing for capacity constraints. The third model aims at designing a capacitated demand-sensitive peak/off-peak timetable. We assessed the performance of these three models and conducted sensitivity analyzes on different parameters on a metro line in Singapore, finding that dynamical timetable built with capacity constraints is most advantageous. Finally, we conclude our study and discuss the implications of the three models: the capacitated model provides a timetable which shows best performance under fixed capacity constraints, while the uncapacitated model may offer optimal temporal train configuration. Although we imposed capacity constraints when designing the optimal peak/off-peak timetable, its performance is not as good as models with dynamical headways. However, it shows advantages such as being easier to operate and more understandable to the passengers.     

Energy-efficient metro train rescheduling with uncertain time-variant passenger demands: An approximate dynamic programming approach

In a heavily congested metro line, unexpected disturbances often occur to cause the delay of the traveling passengers, infeasibility of the current timetable and reduction of the operational efficiency. Due to the uncertain and dynamic characteristics of passenger demands, the commonly used method to recover from disturbances in practice is to change the timetable and rolling stock manually based on the experiences and professional judgements. In this paper, we develop a stochastic programming model for metro train rescheduling problem in order to jointly reduce the time delay of affected passengers, their total traveling time and operational costs of trains. To capture the complexity of passenger traveling characteristics, the arriving ratio of passengers at each station is modeled as a non-homogeneous poisson distribution, in which the intensity function is treated as time-varying origin-to-destination passenger demand matrices. By considering the number of on-board passengers, the total energy usage is modeled as the difference between the tractive energy consumption and the regenerative energy. Then, we design an approximate dynamic programming based algorithm to solve the proposed model, which can obtain a high-quality solution in a short time. Finally, numerical examples with real-world data sets are implemented to verify the effectiveness and robustness of the proposed approaches.     

A hierarchical line planning approach for a large-scale high speed rail network: The China case

Planning a set of train lines in a large-scale high speed rail (HSR) network is typically influenced by issues of longer travel distance, high transport demand, track capacity constraints, and a non-periodic timetable. In this paper, we describe an integrated hierarchical approach to determine line plans by defining the stations and trains according to two classes. Based on a bi-level programming model, heuristics are developed for two consecutive stages corresponding to each classification. The approach determines day-period based train line frequencies as well as a combination of various stopping patterns for a mix of fast trunk line services between major stations and a variety of slower body lines that offer service to intermediate stations, so as to satisfy the predicted passenger transport demand. Efficiencies of the line plans described herein concern passenger travel times, train capacity occupancy, and the number of transfers. Moreover, our heuristics allow for combining many additional conflicting demand–supply factors to design a line plan with predominantly cost-oriented and/or customer-oriented objectives. A range of scenarios are developed to generate three line plans for a real-world example of the HSR network in China using a decision support system. The performance of potential train schedules is evaluated to further examine the feasibility of the obtained line plans through graphical timetables.     

A case study on the coordination of last trains for the Beijing subway network

Passengers may make several transfers between different lines to reach their destinations in urban railway transit networks. Coordination of last trains in feeding lines and connecting lines at transfer stations is especially important because it is the last chance for many travellers to transfer. In this paper, a mathematical method is used to reveal the relationships between passenger transfer connection time (PTCT) and passenger transfer waiting time (PTWT). A last-train network transfer model (LNTM) is established to maximize passenger transfer connection headways (PTCH), which reflect last-train connections and transfer waiting time. Additionally, a genetic algorithm (GA) is developed based upon this LNTM model and used to test a numerical example to verify its effectiveness. Finally, the Beijing subway network is taken as a case study. The results of the numerical example show that the model improves five connections and reduces to zero the number of cases when a feeder train arrives within one headway’s time after the connecting train departed.     

Multiple objective programming for bus operation: A case study for Taipei city

This paper, based on the viewpoints of both the passengers and the operator, uses a multi-objective utility function to formulate an effective bus transportation system. The utility function is composed of six factors of passenger concern including riding time, waiting time, degree of crowdedness, transferring frequencies, standing probability and walking time; and also the two major factors of concern to the bus operator: cost and profit. By using a multi-objective programming technique, a series of noninferior solutions are generated. Also, an application to the Taipei city bus system is presented.     

A practical model for transfer optimization in a transit network: Model formulations and solutions

This paper studies the transit network scheduling problem and aims to minimize the waiting time at transfer stations. First, the problem is formulated as a mixed integer programming model that gives the departure times of vehicles in lines so that passengers can transfer between lines at transfer stations with minimum waiting times. Then, the model is expanded to a second model by considering the extra stopping time of vehicles at transfer stations as a new variable set. By calculating the optimal values for these variables, transfers can be better performed. The sizes of the models, compared with the existing models, are small enough that the models can be solved for small- and medium-sized networks using regular MIP solvers, such as CPLEX. Moreover, a genetic algorithm approach is represented to more easily solve larger networks. A simple network is used to describe the models, and a medium-sized, real-life network is used to compare the proposed models with another existing model in the literature. The results demonstrate significant improvement. Finally, a large-scale, real-life network is used as a case study to evaluate the proposed models and the genetic algorithm approach.     

Hybrid cyclicity: Combining the benefits of cyclic and non-cyclic timetables

We propose a new type of timetable that would combine both the regularity of the cyclic timetables and the flexibility of the non-cyclic ones. In order to do so, several combinations of the two timetables are considered. The regularity is incorporated in their design and the flexibility is evaluated using the passenger satisfaction (in monetary units). Each of the tested timetables is constructed using the Passenger Centric Train Timetabling Problem (PCTTP), that is solved using a simulated annealing heuristic. Note that the PCTTP, unlike the traditional Train Timetabling Problem (TTP), does not take into account the conflicts among trains. The aim of the PCTTP is to design such timetables that the passengers’ satisfaction is maximized and it remains the aim of the TTP to remove any potential conflicts. The performance of each of the considered timetables is assessed on the real network of Israeli Railways. The results of the case study show that our proposed hybrid cyclic timetable can provide the benefits of the cyclic and the non-cyclic timetable simultaneously. This timetable consists of 75% of cyclic trains (securing the regularity of the service) and of 25% of non-cyclic trains (deployed as supplementary trains during the peak hours and capturing the demand fluctuation). The level of the passenger satisfaction of the hybrid cyclic timetable is similar to the level of the non-cyclic one, which has about 18.5% of improvement as compared to the purely cyclic one.     

Dynamic bus holding strategies for schedule reliability: Optimal linear control and performance analysis

As is well known, bus systems are naturally unstable. Without control, buses on a single line tend to bunch, reducing their punctuality in meeting a schedule. Although conventional schedule-based strategies that hold buses at control points can alleviate this problem these methods require too much slack, which slows buses. This delays on-board passengers and increases operating costs.It is shown that dynamic holding strategies based on headways alone cannot help buses adhere to a schedule. Therefore, a family of dynamic holding strategies that use bus arrival deviations from a virtual schedule at the control points is proposed. The virtual schedule is introduced whether the system is run with a published schedule or not. It is shown that with this approach, buses can both closely adhere to a published schedule and maintain regular headways without too much slack.A one-parameter version of the method can be optimized in closed form. This simple method is shown to be near-optimal. To put it in practice, the only data needed in real time are the arrival times of the current bus and the preceding bus at the control point relative to the virtual schedule. The simple method was found to require about 40% less slack than the conventional schedule-based method. When used only to regulate headways it outperforms headway-based methods.