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.     

Long-term planning for ring-radial urban rail transit networks

Extensive work exists on regular rail network planning. However, few studies exist on the planning and design of ring-radial rail transit systems. With more ring transit lines being planned and built in Asia, Europe and the America's, a detailed study on ring transit lines is timely. An analytical model to find the optimal number of radial lines in a city for any demand distribution is first introduced. Secondly, passenger route choice for different rail networks is analyzed, for a many-to-many Origin-Destination (OD) demand distribution, based on a total travel time cost per passenger basis. The routes considered are: (1) radial lines only; (2) ring line only or radial lines and ring line combined; or (3) direct access to a destination without using the rail system. Mathematica and Matlab are used to code the route choice model. A cost-benefit optimization model to identify the feasibility and optimality of a ring line is proposed. Unlike simulations and agent-based models, this model is shown to be easily transferable to many ring-radial transit networks. The City of Calgary is used as an example to illustrate the applicability of each model. The existing urban rail network and trip distribution are major influencing factors in judging the feasibility and optimal location of the ring line. This study shows the potential net benefit of introducing a ring line by assessing changes in passengers’ costs. The changes in passenger cost parameters, such as ride cost and access cost, are shown to greatly influence the feasibility of a ring line.     

Optimal control of headway variation on transit routes

Headway control strategies have been proposed as methods for correcting transit service irregularities and thereby reducing passenger wait times at stops. This paper addresses a particular strategy which can be implemented on high frequency routes (headways under 10–12 minutes), in which buses are held at a control stop to a threshold headway. An algorithm is developed which yields the optimal control stop location and optimal threshold headway with respect to a system wait function. The specification of the wait function is based on the development of several empirical models, including a headway variation model and an average delay time model at control stops. A conclusion is reached that the headway variation does not increase linearly along a route, a common assumption made in many previous studies. Furthermore, the location of the optimal control stop and threshold value are sensitive to the passenger boarding profile, as expected. The algorithm itself appears to have practical application to conventional transit operations.     

Estimation of the perceived value of transit time for containerized cargoes

This paper proposes a novel method for estimating the perceived value of transit time of containers by shipping lines. The key idea is that a shipping line’s published schedule is the optimal decision that minimizes the sum of fuel cost and time-associated costs of the containers adopted by the shipping line. Using the proposed method, we find that the adopted values of transit time for nine trans-Pacific services operated by Orient Overseas Container Line and five trans-Pacific services operated by Maersk Line are between US$5/TEU/day and US$30/TEU/day. We further demonstrate how the adopted value can be used for designing the optimal transit times between ports, analyzing the viability of slow-steaming, checking whether ships should speed up to catch up to connecting ships on other services, and helping to predict the market share of less polluting fuels in view of rules on air emission.     

Stochastic optimal path problem with relays

This paper studies the optimal path problem for travelers driving with vehicles of a limited range, such as most battery electric vehicles currently available in the market. The optimal path in this problem often consists of several relay points, where the vehicles can be refueled to extend its range. We propose a stochastic optimal path problem with relays (SOPPR), which aims at minimizing a general expected cost while maintaining a reasonable arrival probability. To account for uncertainty in the road network, the travel speed on a road segment is treated as a discrete random variable, which determines the total energy required to traverse the segment. SOPPR is formulated in two stages in this paper. In the first stage, an optimal routing problem is solved repeatedly to obtain the expected costs and arrival probabilities from any node to all refueling nodes and the destination. With this information, the second stage constructs an auxiliary network, on which the sequence of refueling decisions can be obtained by solving another optimal path problem. Label-correcting algorithms are developed to solve the routing problems in both stages. Numerical experiments are conducted to compare the stochastic and deterministic models, to examine the impact of different parameters on the routing results, and to evaluate the computational performance of the proposed algorithms.     

A slack arrival strategy to promote flex-route transit services

Flex-route transit, which combines the advantages of fixed-route transit and demand-responsive transit, is one of the most promising options in low-demand areas. This paper proposes a slack arrival strategy to reduce the number of rejected passengers and idle time at checkpoints resulting from uncertain travel demand. This strategy relaxes the departure time constraints of the checkpoints that do not function as transfer stations. A system cost function that includes the vehicle operation cost and customer cost is defined to measure system performance. Theoretical and simulation models are constructed to test the benefits of implementing the slack arrival strategy in flex-route transit under expected and unexpected demand levels. Experiments over a real-life flex-route transit service show that the proposed slack arrival strategy could improve the system performance by up to 40% with no additional operating cost. The results demonstrate that the proposed strategy can help transit operators provide more cost-efficient flex-route transit services in suburban and rural areas.     

Design and development of interactive trip planning for web-based transit information systems

This article presents a Web-based transit information system design that uses Internet Geographic Information Systems (GIS) technologies to integrate Web serving, GIS processing, network analysis and database management. A path finding algorithm for transit network is proposed to handle the special characteristics of transit networks, e.g., time-dependent services, common bus lines on the same street, and non-symmetric routing with respect to an origin/destination pair. The algorithm takes into account the overall level of services and service schedule on a route to determine the shortest path and transfer points. A framework is created to categorize the development of transit information systems on the basis of content and functionality, from simple static schedule display to more sophisticated real time transit information systems. A unique feature of the reported Web-based transit information system is the Internet-GIS based system with an interactive map interface. This enables the user to interact with information on transit routes, schedules, and trip itinerary planning. Some map rendering, querying, and network analysis functions are also provided.     

A cost-competitiveness analysis of charging infrastructure for electric bus operations

This study investigates the cost competitiveness of different types of charging infrastructure, including charging stations, charging lanes (via charging-while-driving technologies) and battery swapping stations, in support of an electric public transit system. To this end, we first establish mathematical models to investigate the optimal deployment of various charging facilities along the transit line and determine the optimal size of the electric bus fleet, as well as their batteries, to minimize total infrastructure and fleet costs while guaranteeing service frequency and satisfying the charging needs of the transit system. We then conduct an empirical analysis utilizing available real-world data. The results suggest that: (1) the service frequency, circulation length, and operating speed of a transit system may have a great impact on the cost competitiveness of different charging infrastructure; (2) charging lanes enabled by currently available inductive wireless charging technology are cost competitive for most of the existing bus rapid transit corridors; (3) swapping stations can yield a lower total cost than charging lanes and charging stations for transit systems with high operating speed and low service frequency; (4) charging stations are cost competitive only for transit systems with very low service frequency and short circulation; and (5) the key to making charging lanes more competitive for transit systems with low service frequency and high operating speed is to reduce their unit-length construction cost or enhance their charging power.     

Measuring the expected locational accessibility of urban transit network for commuting trips

This paper focuses on measuring of the expected locational accessibility (ELA) of urban transit networks for commuters. The ELA of the transit network is measured by a factor named expected locational accessibility index (ELAI), which is calculated based on the expected number of reachable stations with different times of transfers (ENRST) starting from any one transit station on the network. Two approaches, the sample-test-statistics method and the topological analysis method for determining the ENRST are proposed and tested with an example transit network. Finally, the proposed methods are applied to an empirical study for evaluating the ELA performance of bus transit network for commuters of Xiamen City, China. The empirical results show that the ELAI obtained by our two methods are relatively smaller than those obtained by the existing methods. The reason is analyzed to guarantee the accuracy of ELAI measurement.     

Optimal solution to the vehicle routing problem by adopting a meta-heuristic algorithm

The delivery service provided by large-scale retailers continues to grow as online sales occupy an increasingly large share of the market. This study aims to tease out efficient vehicle scheduling times as well as optimal delivery routes by applying meta-heuristic algorithms. Monthly data on existing routes were obtained from a branch of Korea’s leading large-scale online retailer. The first task was to examine the status of existing routes by comparing delivery routes created using Dijkstra’s algorithm with existing delivery routes and their vehicle scheduling. The second task was to identify optimal delivery routes through a comparative analysis of the genetic algorithm and Tabu search algorithm, known for its superior applicability amongst other meta-heuristic algorithms. These findings demonstrate that the optimal vehicle routing problem not only has the potential to reduce distribution costs for operators and expedite delivery for consumers, but also the added social benefit of reduced carbon emissions.     

Bus routing strategies in a transit market: A case study of Hong Kong

Conventionally, the objective of transit routing is often set either to minimize the total operational cost, subject to a given level of service quality, or to maximize the service quality at a given acceptable cost. In a deregulated, commercial‐based environment however, such as bus and railway operations in cities of the UK and Hong Kong where several private firms compete in route‐based or area‐based market, routing becomes one of the means for higher returns rather than just for cost saving. In such a case, how do the transit providers set up their routes for profit‐maximization? Will the routing based on the provider's objective meet the user's objective? How do government regulations and policies affect the choice of transit provider's routing strategy? To answer these questions, we first examine the relationship between the objectives of users and transit providers, set up criteria for transit routing quality, and then investigate the possible routing configurations/patterns for a hypothetical case. These criteria include (1) the load factor of transit, (2) the level of route directness, (3) the level of route overlapping, and (4) the total number of routes and (5) the average of route length. These measures are finally applied to a real case in Hong Kong to examine the route changes of Kowloon Motor Bus from 1975 to 1995. The result of the empirical case reveais how key measures such as load factor are controlled by the bus operator and affected by government policies and how the bus routing pattern was adjusted to meet users' need. Facing the dilemma as evident in Hong Kong between the route directness and the efficiency of road use, we suggest that a rational multi‐modal routing structure be put in place if an institutional solution is introduced so that bus and other transit modes can form a sharing program or an alliance.     

The impact of a financial constraint on the spatial structure of public transport services

Using a single line model, it has been shown recently that the presence of a stringent financial constraint induces a less than optimal bus frequency and larger than optimal bus size. This occurs because the constraint induces a reduction of the importance of users’ costs (their time); in the extreme, users’ costs disappear from the design problem. In this paper we show that such a constraint also has an impact on the spatial structure of transit lines. This is done departing from the single line model using an illustrative urban network that could be served either with direct services (no transfers) or with corridors (transfers are needed). First, the optimal structure of lines is investigated along with frequencies and vehicle sizes when the full costs for users and operators are minimized (unconstrained case); the optimal lines structure is shown to depend upon the patronage level, the values of time and the cost of providing bus capacity. Then the same problem is solved for the extreme case of a stringent financial constraint, in which case users’ costs have relatively little or no effect in determining the solution; in this case the preferred outcome would be direct services under all circumstances, with lower frequencies and larger bus sizes. The impact of the financial constraint on the spatial structure of transit lines is shown to be caused by the reduction in cycle time under direct services; the introduction of users’ costs in the objective function makes waiting times reverse this result under some circumstances.     

Development of a transit network from a street map database with spatial analysis and dynamic segmentation

This paper presents an integrated transit-oriented travel demand modeling procedure within the framework of geographic information systems (GIS). Focusing on transit network development, this paper presents both the procedure and algorithm for automatically generating both link and line data for transit demand modeling from the conventional street network data using spatial analysis and dynamic segmentation. For this purpose, transit stop digitizing, topology and route system building, and the conversion of route and stop data into link and line data sets are performed. Using spatial analysis, such as the functionality to search arcs nearest from a given node, the nearest stops are identified along the associated links of the transit line, while the topological relation between links and line data sets can also be computed using dynamic segmentation. The advantage of this approach is that street map databases represented by a centerline can be directly used along with the existing legacy urban transportation planning systems (UTPS) type travel modeling packages and existing GIS without incurring the additional cost of purchasing a full-blown transportation GIS package. A small test network is adopted to demonstrate the process and the results. The authors anticipate that the procedure set forth in this paper will be useful to many cities and regional transit agencies in their transit demand modeling process within the integrated GIS-based computing environment.     

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.     

An improved headway-based holding strategy for bus transit

Abstract

This paper presents an improved headway-based holding strategy integrating bus transit travel and dwelling time prediction. A support vector machine-based (SVM) model is developed to predict the baseline travel and dwell times of buses based on recent data. In order to reduce prediction errors, an adaptive algorithm is used together with real-time bus operational information and estimated baseline times from SVM models. The objective of the improved holding strategy is to minimize the total waiting times of passengers at the current stop and at successive stops. Considering the time-varying features of bus running, a ‘forgetting factor’ is introduced to weight the most recent data and reduce the disturbance from unexpected incidents. Finally, the improved holding strategy proposed in this study is illustrated using the microscopic simulation model Paramics and some conclusions are drawn.     

A hierarchical customer satisfaction framework for evaluating rail transit systems of Istanbul

This paper provides a hierarchical customer satisfaction framework to measure rail transit lines’ performances in Istanbul. The problems related to rail transit line systems are addressed via customer satisfaction surveys. Then, a framework is proposed combining statistical analysis, fuzzy analytic hierarchy process, trapezoidal fuzzy sets and Choquet integral to evaluate customer satisfaction levels. Next, the criteria need to be improved are determined and specific recommendations to enhance the operation for specific lines are suggested. The proposed framework provides directions for the future investments and it also can be used at a more macroscopic level to determine the operational deficiencies. Furthermore, it can be generalized and applied to complex decision making problems that include uncertain and subjective data or vague information.     

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.     

Mobility evaluation for urban public transportation

Summary

This paper has reported on a study of relative opportunity—not absolute opportunity. Minimum absolute standards for mobility or accessibility are difficult to justify. Some additional study into the development and application of absolute mobility standards may be warranted.

The application of the mobility evaluation model has primarily focused upon a corridor line‐haul system. Conclusions suggest that such a system will not markedly improve existing transit mobility levels in either the peak hour or the off‐peak. The experimental work has verified this conclusion, and more importantly, it has detailed quantitatively the exact levels and spatial distribution of mobility improvements. However, this study does not include a comprehensive analysis of all methods of mobility enhancement, nor does it undertake a comparison of alternative means of mobility improvement. Certainly other methods to improve access to opportunities should be explored before policy considerations are finalized. These methods include other transit solutions, land use alternatives, socio‐economic policies, and other‐mode transportation alternatives. The accessibility technique and mobility indices approach appears to have general applicability in the analysis of optimal strategies for system evaluation.

Of interest is an examination of alternative feeder transit systems to the corridor line. Additional research with the model might point out the maximum mobility effects expected through improved collector service in the suburbs, with corridor line‐haul to the CBD.

The indices are also readily available for a comparison of mobility patterns for different urban areas. Application of the program to transit and socio‐economic data for a set of cities would yield an indication of the relative mobility levels provided. Such data might be considered as an evaluation criterion for future transit funding by federal officials.

In addition, the model is currently being considered by UMTA as a tool to aid in the evaluation of the equitable distribution of transit system benefits as defined in Title VI of the Civil Rights Act of 1964.²⁵ The mobility output would serve as an indicator of the levels‐of‐service provided to certain disadvantaged urban groups. For this application the computer model is being altered to achieve compatability with the Transportation Planning System (UTPS) computer model package developed by UMTA.     

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.