Timetable optimization models and methods for minimizing passenger waiting time at public transit terminals

This paper focuses on developing mathematical optimization models for the train timetabling problem with respect to dynamic travel demand and capacity constraints. The train scheduling models presented in this paper aim to minimize passenger waiting times at public transit terminals. Linear and non-linear formulations of the problem are presented. The non-linear formulation is then improved through introducing service frequency variables. Heuristic rules are suggested and embedded in the improved non-linear formulation to reduce the computational time effort needed to find the upper bound. The effectiveness of the proposed train timetabling models is illustrated through the application to an underground urban rail line in the city of Tehran. The results demonstrate the effectiveness of the proposed demand-oriented train timetabling models, in terms of decreasing passenger waiting times. Compared to the baseline and regular timetables, total waiting time is reduced by 6.36% and 10.55% respectively, through the proposed mathematical optimization models.     

Train design and routing optimization for evaluating criticality of freight railroad infrastructures

Freight transportation by railroads is an integral part of the U.S. economy. Identifying critical rail infrastructures can help stakeholders prioritize protection initiatives or add necessary redundancy to maximize rail network resiliency. The criticality of an infrastructure element, link or yard, is based on the increased cost (delay) incurred when that element is disrupted. An event of disruption can cause heavy congestion so that the capacity at links and yards should be considered when freight is re-routed. This paper proposes an optimization model for making-up and routing of trains in a disruptive situation to minimize the system-wide total cost, including classification time at yards and travel time along links. Train design optimization seeks to determine the optimal number of trains, their routes, and associated blocks, subject to various capacity and operational constraints at rail links and yards. An iterative heuristic algorithm is proposed to attack the computational burden for real-world networks. The solution algorithm considers the impact of volume on travel time in a congested or near-congested network. The proposed heuristics provide quality solutions with high speed, demonstrated by numerical experiments for small instances. A case study is conducted for the network of a major U.S. Class-I railroad based on publicly available data. The paper provides maps showing the criticality of infrastructure in the study area from the viewpoint of strategic planning.     

Robust single-track train dispatching model under a dynamic and stochastic environment: A scenario-based rolling horizon solution approach

After a major service disruption on a single-track rail line, dispatchers need to generate a series of train meet-pass plans at different decision times of the rescheduling stage. The task is to recover the impacted train schedule from the current and future disturbances and minimize the expected additional delay under different forecasted operational conditions. Based on a stochastic programming with recourse framework, this paper incorporates different probabilistic scenarios in the rolling horizon decision process to recognize (1) the input data uncertainty associated with predicted segment running times and segment recovery times and (2) the possibilities of rescheduling decisions after receiving status updates. The proposed model periodically optimizes schedules for a relatively long rolling horizon, while selecting and disseminating a robust meet-pass plan for every roll period. A multi-layer branching solution procedure is developed to systematically generate and select meet-pass plans under different stochastic scenarios. Illustrative examples and numerical experiments are used to demonstrate the importance of robust disruption handling under a dynamic and stochastic environment. In terms of expected total train delay time, our experimental results show that the robust solutions are better than the expected value-based solutions by a range of 10-30%.     

A multiobjective planning model for intercity train seat allocation

This paper presents a multiobjective planning model for generating optimal train seat allocation plans on an intercity rail line serving passengers with many‐to‐many origin‐destination pairs. Two planning objectives of the model are to maximise the operator's total passenger revenue and to minimise the passenger's total discomfort level. For a given set of travel demand, train capacity, and train stop‐schedules, the model is solved by fuzzy mathematical programming to generate a best‐compromise train seat allocation plan. The plan determines how many reserved and non‐reserved seats are to be allocated at each origin station for all subsequent destination stations on each train run operated within a specified operating period. An empirical study on the to‐be‐built Taiwan's high‐speed rail system is conducted to demonstrate the effectiveness of the model. The model can be used for any setting of travel demand and stop‐schedules with various train seating capacities.     

Optimal feeder bus routes on irregular street networks

The methodology presented here seeks to optimize bus routes feeding a major intermodal transit transfer station while considering intersection delays and realistic street networks. A model is developed for finding the optimal bus route location and its operating headway in a heterogeneous service area. The criterion for optimality is the minimum total cost, including supplier and user costs. Irregular and discrete demand distributions, which realistically represent geographic variations in demand, are considered in the proposed model. The optimal headway is derived analytically for an irregularly shaped service area without demand elasticity, with non‐uniformly distributed demand density, and with a many‐to‐one travel pattern. Computer programs are designed to analyze numerical examples, which show that the combinatory type routing problem can be globally optimized. The improved computational efficiency of the near‐optimal algorithm is demonstrated through numerical comparisons to an optimal solution obtained by the exhaustive search (ES) algorithm. The CPU time spent by each algorithm is also compared to demonstrate that the near‐optimal algorithm converges to an acceptable solution significantly faster than the ES algorithm.     

Demand for rail travel to and from airports

Rail access to airports is becoming increasingly important for both train operators and the airports themselves. This paper reports analysis of inter-urban rail demand to and from Manchester and Stansted Airports and the sensitivity of this market segment to growth in air traffic and the cost and service quality of rail services. The estimated demand parameters vary in an expected manner between outward and inward air travellers as well as between airport users and general rail travellers. These parameters can be entered into the demand forecasting framework widely used in the rail industry in Great Britain to provide an appropriate means of forecasting for this otherwise neglected market segment. The novel features of this research, at least in the British context, are that it provides the first detailed analysis of aggregate rail flows to and from airports, it has disaggregated the traditional generalised time measure of rail service quality in order to estimate separate elasticities to journey time, service headway and interchange, and it has successfully explored departures from the conventional constant elasticity position.     

Optimization of grid bus transit systems with elastic demand

Current analytic models for optimizing urban bus transit systems tend to sacrifice geographic realism and detail in order to obtain their solutions. The models presented here shows how an optimization approach can be successful without oversimplifying spatial characteristics and demand patterns of urban areas and how a grid bus transit system in a heterogeneous urban environment with elastic demand is optimized. The demand distribution over the service region is discrete, which can realistically represent geographic variation. Optimal network characteristics (route and station spacings), operating headways and fare are found, which maximize the total operator profit and social welfare. Irregular service regions, many‐to‐many demand patterns, and vehicle capacity constraints are considered in a sequential optimization process. The numerical results show that at the optima the operator profit and social welfare functions are rather flat with respect to route spacing and headway, thus facilitating the tailoring of design variables to the actual street network and particular operating schedule without a substantial decrease in profit. The sensitivities of the design variables to some important exogenous factors are also presented.     

Solving network equilibrium problems on multimodal urban transportation networks with multiple user classes

The capacity of the high‐speed train to compete against travel demand in private vehicles is analysed. A hypothetical context analysed as the high‐speed alternative is not yet available for the route studied. In order to model travel demand, experimental designs were applied to obtain stated preference information. Discrete choice logit models were estimated in order to derive the effect of service variables on journey utility. From these empirical demand models, it was possible to predict for different travel contexts and individuals the capacity of the high‐speed train to compete with the car, so determining the impact of the new alternative on modal distribution. Furthermore, individual willingness to pay for travel time saving is derived for different contexts. The results allow us to confirm that the high‐speed train will have a significant impact on the analysed market, with an important shift of passengers to the new rail service being expected. Different transport policy scenarios are derived. The cost of travel appears to a great extent to be a conditioning variable in the modal choice. These results provide additional evidence for the understanding of private vehicle travel demand.     

Modeling and optimizing urban bus transit considering headway variation for cost and service reliability analysis

Due to the stochastic nature of traffic conditions and demand fluctuations, it is a challenging task for operators to maintain reliable services, and passengers often suffer from longer travel times. A failure to consider this issue while planning bus services may lead to undesirable results, such as higher costs and a deterioration in level of service. Considering headway variation at route stops, this paper develops a mathematical model to optimize bus stops and dispatching headways that minimize total cost, consisting of both user and operator costs. A Genetic Algorithm is applied to search for a cost-effective solution in a real-world case study of a bus transit system, which improves service reliability in terms of a reduced coefficient of variation of headway.     

Joint optimization of a rail transit line and its feeder bus system

A model is developed for jointly optimizing the characteristics of a rail transit route and its associated feeder bus routes in an urban corridor. The corridor demand characteristics are specified with irregular discrete distributions which can realistically represent geographic variations. The total cost (supplier plus user cost) of the integrated bus and rail network is minimized with an efficient iterative method that successively substitutes variable values obtained through classical analytic optimization. The optimized variables include rail line length, rail station spacings, bus headways, bus stop spacings, and bus route spacing. Computer programs are designed for optimization and sensitivity analysis. The sensitivity of the transit service characteristics to various travel time and cost parameters is discussed. Numerical examples are presented for integrated transit systems in which the rail and bus schedules may be coordinated.     

Multi-objective optimization of train routing problem combined with train scheduling on a high-speed railway network

Based on train scheduling, this paper puts forward a multi-objective optimization model for train routing on high-speed railway network, which can offer an important reference for train plan to provide a better service. The model does not only consider the average travel time of trains, but also take the energy consumption and the user satisfaction into account. Based on this model, an improved GA is designed to solve the train routing problem. The simulation results demonstrate that the accurate algorithm is suitable for a small-scale network, while the improved genetic algorithm based on train control (GATC) applies to a large-scale network. Finally, a sensitivity analysis of the parameters is performed to obtain the ideal parameters; a perturbation analysis shows that the proposed method can quickly handle the train disturbance.     

Modeling a rail transit alignment considering different objectives

An optimization model for station locations for an on-ground rail transit line is developed using different objective functions of demand and cost as both influence the planning of a rail transit alignment. A microscopic analysis is performed to develop a rail transit alignment in a given corridor considering a many-to-one travel demand pattern. A variable demand case is considered as it replicates a realistic scenario for planning a rail transit line. A Genetic Algorithm (GA) based on a Geographical Information System (GIS) database is developed to optimize the station locations for a rail transit alignment. The first objective is to minimize the total system cost per person, which is a function of user cost, operator cost, and location cost. The second objective is to maximize the ridership or the service coverage of the rail transit alignment. The user cost per person is minimized separately as the third objective because the user cost is one of the most important decision-making factors for planning a transit system from the users’ perspective. A transit planner can make an informed decision between various alternatives based on the results obtained using different objective functions. The model is applied in a case study in the Washington, DC area. The optimal locations and sequence of stations obtained using the three objective functions are presented and a comparative study between the results obtained is shown in the paper. In future works we will develop a combinatorial optimization problem using the aforementioned objectives for the rail transit alignment planning and design problem.     

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 robust,tactic-based,real-time framework for public-transport transfer synchronization

Missed transfers affect public transport (PT) operations by increasing passenger’s waiting and travel times and frustration. Because of the stochastic and uncertain nature of PT systems, synchronized transfers do not always materialize. This work proposes a new mathematical programming model to minimize total passenger travel time and maximize direct (without waiting) transfers. The model consists of four policies built on a combination of three tactics: holding, skip-stops, and short-turn, the last applied, for the first time, as a real-time control action. The concept is implemented in two steps: optimization and simulation. An agent-based simulation framework is used to represent real-life scenarios, generate random input data, and validate the optimization results. In order to assess the robustness of this framework, a wide range of schedule-deviation scenarios are defined using efficient algorithms for solving the control models within a rolling horizon structure. A case study of the Auckland, New Zealand, PT system is described for assessing the methodology developed. The results show a 4.7% reduction in total passenger travel time and a more than 150% increase in direct transfers. The best impressive results are attained under short headway operations.     

Train scheduling for minimizing passenger waiting time with time-dependent demand and skip-stop patterns: Nonlinear integer programming models with linear constraints

This paper focuses on how to minimize the total passenger waiting time at stations by computing and adjusting train timetables for a rail corridor with given time-varying origin-to-destination passenger demand matrices. Given predetermined train skip-stop patterns, a unified quadratic integer programming model with linear constraints is developed to jointly synchronize effective passenger loading time windows and train arrival and departure times at each station. A set of quadratic and quasi-quadratic objective functions are proposed to precisely formulate the total waiting time under both minute-dependent demand and hour-dependent demand volumes from different origin–destination pairs. We construct mathematically rigorous and algorithmically tractable nonlinear mixed integer programming models for both real-time scheduling and medium-term planning applications. The proposed models are implemented using general purpose high-level optimization solvers, and the model effectiveness is further examined through numerical experiments of real-world rail train timetabling test cases.     

基于遗传算法的地面常规公交与城市轨道交通接驳研究

为了研究地面常规公交与城市轨道接驳问题,构建了基于乘客交通出行时间最短优化模型,采用遗传算法进行求解,并通过具体案例进行了模型验证。结果表明建立的优化模型及遗传算法适用于接驳问题。     

Equity-oriented skip-stopping schedule optimization in an oversaturated urban rail transit network

In this study, we focus on improving system-wide equity performance in an oversaturated urban rail transit network based on multi-commodity flow formulation. From the system perspective, an urban rail transit network is a distributed system, where a set of resources (i.e., train capacity) is shared by a number of users (i.e., passengers), and equitable individuals and groups should receive equal shares of resources. However, when oversaturation occurs in an urban rail transit network during peak hours, passengers waiting at different stations may receive varying shares of train capacity leading to the inequity problem under train all-stopping pattern. Train skip-stopping pattern is an effective operational approach, which holds back some passengers at stations and re-routes their journeys in the time dimension based on the available capacity of each train. In this study, the inequity problem in an oversaturated urban rail transit network is analyzed using a multi-commodity flow modeling framework. In detail, first, discretized states, corresponding to the number of missed trains for passengers, are constructed in a space-time-state three-dimensional network, so that the system-wide equity performance can be viewed as a distribution of all passengers in different states. Different from existing flow-based optimization models, we formulate individual passenger and train stopping pattern as commodity and network structure in the multi-commodity flow-modeling framework, respectively. Then, we aim to find an optimal commodity flow and well-designed network structure through the proposed multi-commodity flow model and simultaneously achieve the equitable distribution of all passengers and the optimal train skip-stopping pattern. To quickly solve the proposed model and find an optimal train skip-stopping pattern with preferable system-wide equity performance, the proposed linear programming model can be effectively decomposed to a least-cost sub-problem with positive arc costs for each individual passenger and a least-cost sub-problem with negative arc costs for each individual train under a Lagrangian relaxation framework. For application and implementation, the proposed train skip-stopping optimization model is applied to a simple case and a real-world case based on Batong Line in the Beijing Subway Network. The simple case demonstrates that our proposed Lagrangian relaxation framework can obtain the approximate optimal solution with a small-gap lower bound and a lot of computing time saved compared with CPLEX solver. The real-world case based on Batong Line in the Beijing Subway Network compares the equity and efficiency indices under the operational approach of train skip-stopping pattern with those under the train all-stopping pattern to state the advantage of the train skip-stopping operational approach.     

Train scheduling simulation that minimises operational conflicts due to service constraints

This paper presents an attempt made to facilitate re‐scheduling of trains to minimize operational delays and accommodate uniform headways for off peak sub urban services subject to resource constraints such as locomotive availability, poor track conditions and stations without siding facilities. The paper describes the computer simulation model designed to optimize train schedules on single‐track rail lines. Using this simulation program it is possible to plan and optimize timetables for railway networks with train runs within short time periods for both single track and double track conditions. The paper describes the capabilities of presenting the results of the simulation runs. These include the time‐distance graph, the network with train movements, dialog boxes with information about selected trains. The programme is capable of changing the starting point, departure time, train destinations and adding or deleting a stop etc. from the user interface. Four objects of array variables are used in the simulation process to keep train and station data. Two object arrays are used for the train movements in up and down directions. The stations' data are stored in the other two object arrays. One of these arrays of stations contains all the stations of the line while the other one contains only the stations with siding facilities. A case study that covers a 61 km long single‐track line with 14 stations is presented to highlight the model capabilities.     

Analyzing the theoretical capacity of railway networks with a radial-backbone topology

In this work we propose a mechanism to optimize the capacity of the main corridor within a railway network with a radial-backbone or X-tree structure. The radial-backbone (or X-tree) structure is composed of two types of lines: the primary lines that travel exclusively on the common backbone (main corridor) and radial lines which, starting from the common backbone, branch out to individual locations. We define possible line configurations as binary strings and propose operators on them for their analysis, yielding an effective algorithm for generating an optimal design and train frequencies. We test our algorithm on real data for the high speed line Madrid–Seville. A frequency plan consistent with the optimal capacity is then proposed in order to eliminate the number of transfers between lines as well as to minimize the network fleet size, determining the minimum number of vehicles needed to serve all travel demand at maximum occupancy.