Multiple period optimization of bus transit systems

Analytic models are developed for optimizing bus services with time dependence and elasticity in their demand characteristics. Some supply parameters, i.e. vehicle operating costs and speeds are also allowed to vary over time. The multiple period models presented here allow some of the optimized system characteristics (e.g. route structure) to be fized at values representing the best compromise over different time periods, while other characteristics (e.g. service headways) may be optimized within each period. In a numerical example the demand is assumed to fluctuate over a daily cycle (e.g. peak, offpeak and night), although the same models can also be used for other cyclical or noncyclical demand variations over any number of periods. Models are formulated and compared for four types of conditions, which include steady fixed demand, cyclical fixed demand, steady equilibrium demand and cyclical equilibrium demand. When fixed demand is assumed, the optimization objective is minimum total system cost, including operator cost and user cost, while operator profit and social welfare are the objective functions maximized for equilibrium demand. The major results consist of closed form solutions for the route spacings, headways, fares and costs for optimized feeder bus services under various demand conditions. A comparison of the optimization results for the four cases is also presented. When demand and bus operating characteristics are allowed to vary over time, the optimal functions are quite similar to those for steady demand and supply conditions. The optimality of a constant ratio between the headway and route spacing, which is found at all demand densities if demand is steady, is also maintained with a multi-period adjustment factor in cyclical demand cases, either exactly or with a relatively negligible approximation. These models may be used to analyze and optimize fairly complex feeder or radial bus systems whose demand and supply characteristics may vary arbitrarily over time.     

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.     

Welfare maximization for bus transit systems with timed transfers and financial constraints

Conventional and flexible bus services may be combined to better serve regions with a wide range of characteristics. If demand densities and resulting service frequencies are low, the coordination of bus arrivals at transfer stations may significantly reduce passenger transfer times. A method is proposed for integrating, coordinating, and optimizing bus services while considering many‐to‐many travel patterns, demand elasticity, financial constraints, and appropriate service type for various regions. The objective is to maximize welfare, that is, the sum of producer and consumer surplus. The problem is solved with a hybrid optimization method, in which a genetic algorithm with bounded integer variables is selected for solving one of the subproblems. The service types, fares, headways, and service zone sizes are jointly optimized. Sensitivity analyses explore how the choice among conventional and flexible busses depends on the demand, subsidy, and demand elasticity parameters. The results also show that welfare can increase due to coordination, and these increases are found to be higher in cases with high demand or low subsidy. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimizing sustainable feeder bus operation considering realistic networks and heterogeneous demand

A mathematical model is developed to optimize social and fiscal sustainable operation of a feeder bus system considering realistic network and heterogeneous demand. The objective total profit is a nonlinear, mixed integer function, which is maximized by optimizing the number of stops, headway, and fare. The stops are located which maximize the ridership. The demand elasticity for the bus service is dependent on passengers' access distance, wait time, in‐vehicle time, and fare. An optimization algorithm is developed to search for the optimal solution that maximizes the profit. The modeling approach is applied to planning a bus transit system within Woodbridge, New Jersey. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

Reducing the cost of peak hour transit service through contracting out service

Transit service contracting has responded to fiscal and financial woes of public transit agencies as the most uniquely attractive cost‐saving strategy at present. Most transit service contracting, however, has been in the traditional provision of entire fixed route bus service or commuter express bus service, and exclusive demand responsive service for the general public or for special disadvantaged population groups such as the elderly and/or the handicapped. This paper presents a new module in transit service contracting whereby the public and private operators jointly provide the peak service on the same route and at the same time. While the public agency provides the base demand of the service, the private provider provides the excess demand, both following the same schedules and similar service arrangements. In this paper, proposed service arrangements, costing and contracting procedures are discussed. It is also reported that substantial cost savings ranging from 32 to 57% with an average savings of 48% can be achieved if the excess peak hour bus transit service on highly peaked routes in public transit agencies is contracted to competing private operator(s).     

A systematic analysis of multimodal transport systems with road space distribution and responsive bus service

A smart design of transport systems involves efficient use and allocation of the limited urban road capacity in the multimodal environment. This paper intends to understand the system-wide effect of dividing the road space to the private and public transport modes and how the public transport service provider responds to the space changes. To this end, the bimodal dynamic user equilibrium is formulated for separated road space. The Macroscopic Fundamental Diagram (MFD) model is employed to depict the dynamics of the automobile traffic for its state-dependent feature, its inclusion of hypercongestion, and its advantage of capturing network topology. The delay of a bus trip depends on the running speed which is in turn affected by bus lane capacity and ridership. Within the proposed bimodal framework, the steady-state equilibrium traffic characteristics and the optimal bus fare and service frequency are analytically derived. The counter-intuitive properties of traffic condition, modal split, and behavior of bus operator in the hypercongestion are identified. To understand the interaction between the transport authority (for system benefit maximization) and the bus operator (for its own benefit maximization), we examine how the bus operator responds to space changes and how the system benefit is influenced with the road space allocation. With responsive bus service, the condition, under which expanding bus lane capacity is beneficial to the system as a whole, has been analytically established. Then the model is applied to the dynamic framework where the space allocation changes with varying demand and demand-responsive bus service. We compare the optimal bus services under different economic objectives, evaluate the system performance of the bimodal network, and explore the dynamic space allocation strategy for the sake of social welfare maximization.     

Estimation of urban bus transit marginal cost without cost data

We develop a method to study the industrial structure of urban bus transit without using cost data. To do so, we estimate the marginal cost function under the assumption that firms compete on frequency and adjust frequency to maximize profits. Our methodology is applied to Santiago, Chile. In this case, demand is modeled with a simplified model of transit network assignment. The goal is to consider how frequency, capacity, and on-board passengers affect the bus line’s demand. The marginal cost function is estimated by using the first-order conditions of the firm’s profit maximization problem, using the results of the demand model as data. We conclude that the urban bus transit industry in Santiago exhibits increasing returns to scale for low levels of demand and that these returns are exhausted rapidly at a moderate demand level. Additionally, firms exhibit economies of network expansion, on average.     

Influential vectors in fuel consumption by an urban bus operator: Bus route,driver behavior or vehicle type?

Energy costs account for an important share of the total costs of urban and suburban bus operators. The purpose of this paper is to expand empirical research on bus transit operation costs and identify the key factors that influence bus energy efficiency of the overall bus fleet of one operator and aid to the management of its resources.We estimate a set of multivariate regression models, using cross-section dataset of 488 bus drivers operating over 92 days in 2010, in 87 routes with different bus typologies, of a transit company operating in the Lisbon’s Metropolitan Area (LMA), Rodoviária de Lisboa, S.A.Our results confirm the existence of influential variables regarding energy efficiency and these are mainly: vehicle type, commercial speed, road grades over 5% and bus routes; and to a lesser extent driving events such as: sudden longitudinal decelerations and excessive engine rotation. The methodology proved to be useful for the bus operator as a decision-support tool for efficiency optimization purpose at the company level.     

Flexible feeder transit route design to enhance service accessibility in urban area

To improve the accessibility of transit system in urban areas, this paper presents a flexible feeder transit routing model that can serve irregular‐shaped networks. By integrating the cost efficiency of fixed‐route transit system and the flexibility of demand responsive transit system, the proposed model is capable of letting operating feeder busses temporarily deviate from their current route so as to serve the reported demand locations. With an objective of minimizing total bus travel time, a new operational mode is then proposed to allow busses to serve passengers on both street sides. In addition, when multiple feeder busses are operating in the target service area, the proposed model can provide an optimal plan to locate the nearest one to response to the demands. A three‐stage solution algorithm is also developed to yield meta‐optimal solutions to the problem in a reasonable amount of time by transforming the problem into a traveling salesman problem. Numerical studies have demonstrated the effectiveness of the proposed model as well as the heuristic solution approach. Copyright © 2015 John Wiley & Sons, Ltd.     

A simulation‐based reliability assessment approach for congested transit network

This paper is an attempt to develop a generic simulation‐based approach to assess transit service reliability, taking into account interaction between network performance and passengers' route choice behaviour. Three types of reliability, say, system wide travel time reliability, schedule reliability and direct boarding waiting‐time reliability are defined from perspectives of the community or transit administration, the operator and passengers. A Monte Carlo simulation approach with a stochastic user equilibrium transit assignment model embedded is proposed to quantify these three reliability measures of transit service. A simple transit network with a bus rapid transit (BRT) corridor is analysed as a case study where the impacts of BRT components on transit service reliability are evaluated preliminarily.     

The Downs–Thomson paradox with imperfect mode substitutes and alternative transit administration regimes

The Downs–Thomson paradox (D–T paradox) occurs when expansion of a congested and untolled highway undermines scale economies of a competing transit service, leaving users of both modes worse off. The standard analysis of the D–T paradox is based on several stringent assumptions: fixed total travel demand, perfect substitutability between automobile and transit trips, and no transit crowding. This paper re-examines the paradox when these assumptions are relaxed while retaining the usual assumption that there is no congestion interaction between the modes. It also broadens consideration to alternative transit administration regimes. In the standard treatment the transit operator is obliged to cover its costs. In this paper we also study two other regimes: transit profit maximization, and system-wide welfare maximization with no financing constraint. We examine how the transit system operator responds to highway capacity expansion in each regime, and how this affects welfare for drivers and transit users. We show that in all regimes the full price of transit declines only if the full price of driving falls as well. Thus, drivers are more likely to benefit from highway expansion than transit riders. The D–T paradox cannot occur in the profit maximization or unconstrained welfare maximization regimes. In the traditional self-financing regime transit service deteriorates, but the D–T paradox is not inevitable. Numerical analysis suggests that it can occur only when automobile and transit trips are nearly perfect substitutes.     

Integrating geographic information systems into transit ridership forecast models

Researchers have produced sophisticated modal split and transit demand models, including forecasts that are sensitive to the level of service. However, little effort has been made to integrate these models into corridor studies and route alignment analyses since (a) re-routing is itself an extremely complex modeling task, and (b) the results of the demand models are presented in tabular form with no facility to visualize spatial patterns and relationships that, if recognized, would aid in the routing tasks. GIS tools can be used, together with the demand models, to identify both clusters of city blocks that house families with certain socioeconomic characteristics and potential trip destinations conducive to transit use. In other words, GIS tools can be used to better measure some of the factors that are needed by transit demand models. The results of these models can be displayed graphically, enabling analysts to target places needing improved service, evaluate route re-alignment alternatives, and operate more efficient and effective bus lines. This paper examines how a particular class of model used by transit agencies for estimating ridership can be integrated with GIS tools in order to facilitate such analyses. It also explores the effects of visualization of routes, demographics, and employment data on the process of designing route alignments with better targeting of high transit ridership areas. This paper is part of a research project sponsored by the Region One University Transportation Center, at MIT.     

Maximizing net benefits for conventional and flexible bus services

Transit ridership is usually sensitive to fares, travel times, waiting times, and access times, among other factors. Therefore, the elasticities of demand with respect to such factors should be considered in modeling bus transit services and must be considered when maximizing net benefits (i.e. “system welfare” = consumer surplus + producer surplus) rather just minimizing costs. In this paper welfare is maximized with elastic demand relations for both conventional (fixed route) and flexible-route services in systems with multiple dissimilar regions and periods. As maximum welfare formulations are usually too complex for exact solutions, they have only been used in a few studies focused on conventional transit services. This limitation is overcome here for both conventional and flexible transit services by using a Real Coded Genetic Algorithm to solve such mixed integer nonlinear welfare maximization problems with constraints on capacities and subsidies. The optimized variables include service type, zone sizes, headways and fares. We also determine the maximum welfare threshold between optimized conventional and flexible services) and explore the effects of subsidies. The proposed planning models should be useful in selecting the service type and optimizing other service characteristics based on local geographic characteristics and financial constraints.     

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.     

Secrets of success: assessing the large increases in transit ridership achieved by Houston and San Diego transit providers

This paper summarizes and updates the findings from an earlier study by the same authors of transit systems in Houston (all bus) and San Diego (bus and light rail). Both systems achieved unusually large increases in transit ridership during a period in which most transit systems in other metropolitan areas were experiencing large losses. Based on ridership models estimated using cross section and time series data, the paper quantifies the relative contributions of policy variables and factors beyond the control of transit operators on ridership growth. It is found that large ridership increases in both areas are caused principally by large service increases and fare reductions, as well as metropolitan employment and population growth. In addition, the paper provides careful estimates of total and operating costs per passenger boarding and per passenger mile for Houston's bus operator and San Diego's bus and light rail operators. These estimates suggest that the bus systems are more cost-effective than the light rail system on the basis of total costs. Finally, the paper carries out a series of policy simulations to analyze the effects of transit funding levels and metropolitan development patterns on transit ridership and farebox recovery ratio.     

Radial bus networks with multi-period and many-to-many demand

Mathematical models are developed for optimizing radial bus networks with time dependent demand and supply characteristics. These models can deal with many-to-many demand distribution in heterogeneous rather than idealized geographic environments. With some approximations, closed-form solutions for the optimal route angle, headways for different time periods, and stop spacings for various locations are obtained for a total cost minimization objective. The relations between the decision variables and system parameters are identified analytically. The optimality of a constant ratio between headways and route angle is found to hold with a time related factor. The optimized wait cost, operator cost, and lateral access cost are found to be equal. A numerical example is given for a case with three service periods. It illustrates the applicability of the analytic model to irregular demand patterns that may be directionally imbalanced during some periods.     

Impact of congestion charging on the transit market: An inter-modal equilibrium model

An inter-modal equilibrium model links an urban road network subject to a congestion charge to a parallel urban transit market, with a view to finding the optimum congestion charge consistent with the commercial decisions of the transit operator(s). A congestion charge is set to maximise social surplus. Travel behaviour is assumed to conform to elastic-demand user equilibrium traffic assignment. The transit market is assumed to be either a profit maximising monopoly or a profit maximising duopoly competing non-cooperatively. The operator(s) set the fares to maximise profits and the supply of transit services are determined by the resulting demand. The problem has been formulated as a bi-level programme with the determination of the congestion charge on the upper level and the setting of transit fares on the lower level. In the case of non-cooperating operators, the Bertrand–Nash equilibrium fares are sought. The results of the model are analysed for a small example based loosely on Edinburgh. This reveals the importance of competition in the transit market for the trade off between the government, the transit provider(s) and the travellers.     

Intermodal Transit System Coordination

In urban areas where transit demand is widely spread, passengers may be served by an intermodal transit system, consisting of a rail transit line (or a bus rapid transit route) and a number of feeder routes connecting at different transfer stations. In such a system, passengers may need one or more transfers to complete their journey. Therefore, scheduling vehicles operating in the system with special attention to reduce transfer time can contribute significantly to service quality improvements. Schedule synchronization may significantly reduce transfer delays at transfer stations where various routes interconnect. Since vehicle arrivals are stochastic, slack time allowances in vehicle schedules may be desirable to reduce the probability of missed connections. An objective total cost function, including supplier and user costs, is formulated for optimizing the coordination of a general intermodal transit network. A four-stage procedure is developed for determining the optimal coordination status among routes at every transfer station. Considering stochastic feeder vehicle arrivals at transfer stations, the slack times of coordinated routes are optimized, by balancing the savings from transfer delays and additional cost from slack delays and operating costs. The model thus developed is used to optimize the coordination of an intermodal transit network, while the impact of a range of factors on coordination (e.g., demand, standard deviation of vehicle arrival times, etc) is examined.     

A GIS‐based method to identify cost‐effective routes for rural deviated fixed route transit

Deviated fixed route transit (DFRT) service connecting rural and urban areas is a growing transportation mode in the USA. Little research has been done to develop frameworks for route design. A methodology to explore the most cost‐effective DFRT route is presented in this paper. The inputs include potential DFRT demand distribution and a road network. A heuristic is used to build possible routes by starting at urban cores and extending in all network directions in certain length increments. All the DFRT routes falling in the length range desired by the users are selected. The cost effectiveness of those routes, defined by operating cost per passenger trip, is compared. The most cost‐effective route is selected and presented in a GIS map. A case study illustrates the methodology in several Tennessee metropolitan regions. The most cost‐effective route length is case specific; some routes (e.g. those out of our Nashville case) are most cost effective when short, while others (e.g. those out of Memphis) are most cost effective when long. Government agencies could use the method to identify routes with the lowest operating cost per passenger given a route length or an operating cost budget. Copyright © 2016 John Wiley & Sons, Ltd.