Techniques for inserting additional trains into existing timetables

In this paper techniques for scheduling additional train services (SATS) are considered as is train scheduling involving general time window constraints, fixed operations, maintenance activities and periods of section unavailability. The SATS problem is important because additional services must often be given access to the railway and subsequently integrated into current timetables. The SATS problem therefore considers the competition for railway infrastructure between new services and existing services belonging to the same or different operators. The SATS problem is characterised as a hybrid job shop scheduling problem with time window constraints. To solve this problem constructive algorithm and meta-heuristic scheduling techniques that operate upon a disjunctive graph model of train operations are utilised. From numerical investigations the proposed framework and associated techniques are tested and shown to be effective.     

Optimal timetable development for community shuttle network with metro stations

This paper investigates an issue for optimizing synchronized timetable for community shuttles linked with metro service. Considering a passenger arrival distribution, the problem is formulated to optimize timetables for multiple community shuttle routes, with the objective of minimizing passenger’s schedule delay cost and transfer cost. Two constraints, i.e., vehicle capacity and fleet size, are modeled in this paper. The first constraint is treated as soft, and the latter one is handled by a proposed timetable generating method. Two algorithms are employed to solve the problem, i.e., a genetic algorithm (GA) and a Frank–Wolfe algorithm combined with a heuristic algorithm of shifting departure times (FW-SDT). FW-SDT is an algorithm specially designed for this problem. The simulated and real-life examples confirm the feasibility of the two algorithms, and demonstrate that FW-SDT outperforms GA in both accuracy and effectiveness.     

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.     

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.     

Scheduling and platforming trains at busy complex stations

We consider the problem of train planning or scheduling for large, busy, complex train stations, which are common in Europe and elsewhere, though not in North America. We develop the constraints and objectives for this problem, but these are too computationally complex to solve by standard combinatorial search or integer programming methods. Also, the problem is somewhat political in nature, that is, it does not have a clear objective function because it involves multiple train operators with conflicting interests. We therefore develop scheduling heuristics analogous to those successfully adopted by train planners using “manual” methods. We tested the model and algorithms by applying to a typical large station that exhibits most of the complexities found in practice. The results compare well with those found by traditional methods, and take account of cost and preference trade-offs not handled by those methods. With successive refinements, the algorithm eventually took only a few seconds to run, the time depending on the version of the algorithm and the scheduling problem. The scheduling models and algorithms developed and tested here can be used on their own, or as key components for a more general system for train scheduling for a rail line or network.Train scheduling for a busy station includes ensuring that there are no conflicts between several hundred trains per day going in and out of the station on intersecting paths from multiple in-lines and out-lines to multiple platforms, while ensuring that each train is allowed at least its minimum required headways, dwell time, turnaround time and trip time. This has to be done while minimizing (costs of) deviations from desired times, platforms or lines, allowing for conflicts due to through-platforms, dead-end platforms, multiple sub-platforms, and possible constraints due to infrastructure, safety or business policy.     

Optimizing the demand captured by a railway system with a regular timetable

The railway systems in various European countries adopt regular timetables, in which the trains arrive and depart at constant intervals. In fact, their simple structure provides several advantages both to the passengers and to the management of the service. The design of such timetables has recently received a certain attention in the literature, but the standard model aims to optimize the service for a fixed demand. We relax this unrealistic assumption, taking into account the reciprocal influence between the quality of the timetable and the amount of transport demand captured by the railway. This results into a mixed-integer non linear model with a non-convex continuous relaxation. We solve it by a branch-and-bound algorithm based on a piecewise-linear overestimate of the objective function and a heuristic algorithm which iteratively applies the standard fixed-demand model and a demand-estimation model, feeding each one with data based on the solution obtained from the other one at the previous iteration. The computational results presented concern both random instances and a real-world regional network located in Northwestern Italy.     

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.     

An appraisal of a column-generation-based algorithm for centralized train-conflict resolution on a metropolitan railway network

In practice, a train-conflict resolution is decentralized around dispatchers each of whom controls a few segments in a global railway network with her rule-of-thumb to operational data. Conceptually, the global sub-optimality or infeasibility of the decentralized system is resolved by a network controller who coordinates the dispatchers and train operators at the lower layers on a real-time basis. However, such notion of a multi-layer system cannot be effectual unless the top layer is able to provide a global solution soon enough for the dynamic lower layers to adapt in a seamless manner. Unfortunately, a train-conflict resolution problem is NP-hard as formally established in this paper and an effective solution method traded off between computation time and solution quality has been lacking in literature. Thus, we propose a column-generation-based algorithm that exploits the separability of the problem. A key ingredient of the algorithm is an efficient heuristic for the pricing subproblem for column generation. Tested on the real data from the Seoul metropolitan railway network, the algorithm provides near-optimal conflict-free timetables in a few seconds for most cases. The performance of the proposed algorithm is compared to the ones of the previous MIP-based heuristic by Törnquist and Persson (2007) and the priority-based heuristic by Sahin (1999).     

A schedule‐based dynamic transit network model — Recent advances and prospective future research

Using the schedule‐based approach, in which scheduled timetables are used to describe the movement of vehicles, a dynamic transit assignment model is formulated. Passengers are assumed to travel on a path with minimum generalized cost that consists of four components: in‐vehicle time; waiting time; walking time; and a time penalty for each line change. A specially developed branch and bound algorithm is used to generate the time‐dependent minimum path. The assignment procedure is conducted over a period in which both passenger demand and train headway are varying. This paper presents an overview of the research that has been carried out by the authors to develop the schedule‐based transit assignment model, and offers perspectives for future research.     

Designing an integrated distribution system for catering services for high-speed railways: A three-echelon location routing model with tight time windows and time deadlines

An emerging task in catering services for high-speed railways (CSHR) is to design a distribution system for the delivery of high-quality perishable food products to trains in need. This paper proposes a novel model for integrating location decision making with daily rail catering operations, which are affected by various aspects of rail planning, to meet time-sensitive passenger demands. A three-echelon location routing problem with time windows and time budget constraints (3E-LRPTWTBC) is thus proposed toward formulating this integrated distribution system design problem. This model attempts to determine the capacities/locations of distribution centers and to optimize the number of meals delivered to stations. The model also attempts to generate a schedule for refrigerated cars traveling from distribution centers to rail stations for train loading whereby meals can be catered to trains within tight time windows and sold before a specified time deadline. By relaxing the time-window constraints, a relaxation model that can be solved using an off-the-shelf mixed integer programming (MIP) solver is obtained to provide a lower bound on the 3E-LRPTWTBC. A hybrid cross entropy algorithm (HCEA) is proposed to solve the 3E-LRPTWTBC. A small-scale case study is implemented, which reveals a 9.3% gap between the solution obtained using the HCEA and that obtained using the relaxation model (RM). A comparative analysis of the HCEA and an exhaustive enumeration algorithm indicates that the HCEA shows good performance in terms of computation time. Finally, a case study considering 156 trains on the Beijing-Shanghai high-speed corridor and a large-scale case study considering 1130 trains on the Chinese railway network are addressed in a comprehensive study to demonstrate the applicability of the proposed models and algorithm.     

A bi‐level programming approach — Optimal transit fare under line capacity constraints

The fare of a transit line is one of the important decision variables for transit network design. It has been advocated as an efficient means of coordinating the transit passenger flows and of alleviating congestion in the transit network. This paper shows how transit fare can be optimized so as to balance the passenger flow on the transit network and to reduce the overload delays of passengers at transit stops. A bi‐level programming method is developed to optimize the transit fare under line capacity constraints. The upper‐level problem seeks to minimize the total network travel time, while the lower‐level problem is a stochastic user equilibrium transit assignment model with line capacity constraints. A heuristic solution algorithm based on sensitivity analysis is proposed. Numerical example is used to illustrate the application of the proposed model and solution algorithm.     

A distributed VNS algorithm for optimizing dial-a-ride problems in large-scale scenarios

These days, transportation and logistic problems in large cities are demanding smarter transportation services that provide flexibility and adaptability. A possible solution to this arising problem is to compute the best routes for each new scenario. In this problem, known in the literature as the dial-a-ride problem, a number of passengers are transported between pickup and delivery locations trying to minimize the routing costs while respecting a set of prespecified constraints. This problem has been solved in the literature with several approaches from small to medium sized problems. However, few efforts have dealt with large scale problems very common in massive scenarios (big cities or highly-populated regions). In this study, a new distributed algorithm based on the partition of the requests space and the combination of the routes is presented and tested on a set of 24 different scenarios of a large-scale problem (up to 16,000 requests or 32,000 locations) in the city of San Francisco. The results show that, not only the distributed algorithm is able to solve large problem instances that the corresponding sequential algorithm is unable to solve in a reasonable time, but also to have an average improvement of 9% in the smaller problems. The results have been validated by means of statistical procedures proving that the distributed algorithm can be an effective way to solve high dimensional dial-a-ride problems.     

Timetable-based operation in urban transport: Run-time optimisation and improvements in the operating process

Urban public transit provides an efficient means of mobility and helps support social development and environmental preservation. To avoid loss of ridership, transit authorities have focussed on improving the punctuality of routes that operate using timetables. This paper presents a new approach to generating run-time values that is based on analytical development and micro simulations. The work utilizes previous research (described herein) and the experience acquired by Transports Metropolitans de Barcelona (TMB) in operating bus routes based on timetables. Using a sample of historical data, the method used for generating run-time values consists of the following steps: purging and screening atypical trips, based on the consideration of confidence intervals for median trips; segmenting the day into time bands based on the introduction of a new hierarchical classification algorithm; creating initial run-time values based on criteria derived from statistical analysis; adjusting and validating initial run-time values using micro simulations; and evaluating incident-recovery times at the end of trips in order to guarantee the punctual departure of the next trip in the vehicle schedule. To favour service improvement, we also introduced certain indicators that can identify the root causes of non-compliance. As a final step, in order to ensure the applicability and use of the model, we promoted the development of our model within the framework of the HASTUS™ software solution.     

Combining multicriteria analysis and tabu search for dial-a-ride problems

In the Dial-a-Ride Problem (DARP) the aim is to design vehicle routes for a set of users who must be transported between given origin and destination pairs, subject to a variety of side constraints. The standard DARP objective is cost minimization. In addition to cost, the objectives considered in this paper include three terms related to quality of service. This gives rise to a multicriteria problem. The problem is solved by means of a flexible and simple metaheuristic which efficiently integrates the reference point method for multicriteria optimization within a tabu search mechanism. Extensive tests were performed on randomly generated data and on real-life data provided by a major transporter in the Montreal area. Results indicate that the algorithm can yield a rich set of non-dominated solutions. It can also be employed to determine good trade-offs between cost and quality of service.     

A differential evolution approach for the vehicle routing problem with backhauls and time windows

This paper presents a differential evolution algorithm (DEA) to solve a vehicle routing problem with backhauls and time windows (VRPBTW) and applied for a catering firm. VRPBTW is an extension of the vehicle routing problem, which includes capacity and time window constraints. In this problem, customers are divided into two subsets: linehaul and backhaul. Each vehicle starts from a depot and goods are delivered from the depot to the linehaul customers. Goods are subsequently brought back to the depot from the backhaul customers. The objective is to minimize the total distance that satisfies all of the constraints. The problem is formulated using mixed integer programming and solved using DEA. Proposed algorithm is tested with several benchmark problems to demonstrate effectiveness and efficiency of the algorithm and results show that our proposed algorithm can find superior solutions for most of the problems in comparison with the best known solutions. Hence, DEA was carried out for catering firm to minimize total transportation costs. Copyright © 2013 John Wiley & Sons, Ltd.     

Service type assignment and container routing with transit time constraints and empty container repositioning for liner shipping service networks

A decision tool is developed for a liner shipping company to deploy its fleet considering vessel speeds and to find routes for cargos with repositioning of empty containers and transit time constraints. This problem is referred as the simultaneous Service type Assignment and container Routing Problem (SARP) in the sequel. A path-flow based mixed-integer linear programming formulation is suggested for the SARP. A Branch and Bound (BB) algorithm is used to solve the SARP exactly. A Column Generation (CG) procedure, embedded within the BB framework, is devised to solve the linear programming relaxation of the SARP. The CG subproblems arises as Shortest Path Problems (SPP). Yet incorporating transit time requirements yields constrained SPP which is NP-hard and solved by a label correcting algorithm. Computational experiments are performed on randomly generated test instances mimicking real life. The BB algorithm yields promising solutions for the SARP. The SARP with and without transit time constraints is compared with each other. Our results suggest a potential to increase profit margins of liner shipping companies by considering transit time requirements of cargos.     

Estimation method for railway passengers’ train choice behavior with smart card transaction data

Smart card systems have become the predominant method of collecting public transport fares in Japan. Transaction data obtained through smart cards have resulted in a large amount of archived information on how passengers use public transportation. The data have the potential to be used for modeling passenger behavior and demand for public transportation. This study focused on train choices made by railway passengers. If each passenger’s train choice can be identified over a long period of time, this information would be useful for improving the customer relationship management of the railway company and for improving train timetables. The aim of this study was to develop a methodology for estimating which train is boarded by each smart card holder. This paper presents a methodology and an algorithm for estimation using long-term transaction data. To validate the computation time and accuracy of the estimation, an empirical analysis is carried out using actual transaction data provided by a railway company in Japan. The results show that the proposed method is capable of estimating passenger usage patterns from smart card transaction data collected over a long time period.     

Fleet size determination for a truckload distribution center

This work presents an analysis of the fleet size determination problem for a truckload distribution center in which each customer requests multi‐truckload services. A transformation is made to define this problem as a cutting stock problem which can be resolved by finding the proper set of patterns beneficial for an NP‐hard (non‐deterministic polynomial‐time hard) problem. The genetic algorithm (GA) is combined with cutting stock techniques to solve the problem. Experiments are carried out testing two types of problems, with and without time‐window constraints. To verify the solution quality obtained with the proposed approach, the problem is also solved using only GA. It is shown that the proposed approach can significantly improve the GA solution by 14.68%. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Optimization of periodic crew schedules with application of column generation method

We present an alternative approach to the problem of periodic crew scheduling. We introduce the concept of frames which leads us to a modeling approach which suits well the current practice of the majority of European railway operators. It results in a model facilitating column generation techniques resulting in a Dantzig-Wolfe type decomposition, and thus suitable for a parallel implementation in a high-performance computing environment. We exploit the properties of network flow models to avoid several additional integer constraints. We compare two approaches to solve the problem. The first approach consists of solving the original problem by single model. The second approach is our step-by-step column generation. The comparison is based on our implementation which we describe in detail along with its application to certain benchmark instances. The benchmarks originate in real or close-to-realistic problems from railway systems in Slovakia and Hungary. The case studies demonstrate that our model is well-suited for real-life applications.