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.     

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.     

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.     

Optimization of headways with stop‐skipping control: a case study of bus rapid transit system

Bus rapid transit system is designed to provide high‐quality and cost‐efficient passenger transportation services. In order to achieve this design objective, effective scheduling strategies are required. This research aims at improving the operation efficiency and service quality of a BRT system through integrated optimization of its service headways and stop‐skipping strategy. Based on cost analysis for both passengers and operation agencies, an optimization model is established. A genetic algorithms based algorithm and an application‐oriented solution method are developed. Beijing BRT Line 2 has been chosen as a case study, and the effectiveness of the optimal headways with stop‐skipping services under different demand levels has been analyzed. The results has shown that, at a certain demand level, the proposed operating strategy can be most advantageous for passengers with an accepted increase of operating costs, under which the optimum headway is between 3.5 and 5.5 min for stop‐skipping services during the morning peak hour depending on the demand with the provision of stop‐skipping services. The effectiveness of the optimal headways with stop‐skipping services is compared with those of existing headways and optimal headways without stop‐skipping services. The results show that operating strategies under the optimal headways with stop‐skipping services outperforms the other two operating strategies with respect to total costs and in‐vehicle time for passengers. Copyright © 2014 John Wiley & Sons, Ltd.     

Scheduling considerations for a branching transit route

This paper addresses the impacts of different scheduling alternatives for a branching transit route. It examines different schedule alternatives that might be used to optimize the route performance in terms of the passenger traveling time distributed among branch passengers and trunk‐line passengers. The schedule alternatives considered include transit vehicle allocation to different branches, offset shifting across vehicles on different branches, and vehicle holding (slack time) in the transit vehicle schedule. With these variables, several vehicle schedules are devised and examined based on a wide variety of possible passenger boarding scenarios using deterministic service models. Test outcomes provide general conclusions about the performance of the strategies. Vehicle assignment leading to even headways among branches is generally preferred for the case of low passenger demand. However, when passenger demand is high, or the differences between the passenger demands on branches are significant, unequal vehicle assignment will be helpful to improve the overall route performance. Holding, as a proactive strategy in scheduling, has the potential to be embedded into the schedule as a type of slack time, but needs further evidence and study to determine the full set of conditions where it may be beneficial. Offset shifting does not show sufficient evidence to be an efficient strategy to improve route performance in the case of low or high passenger demand.     

Integrated transit services for minimum cost operation considering heterogeneous demand

Abstract

In large metropolitan areas, public transit is a major mode choice of commuters for their daily travel, which has an important role in relieving congestion on transportation corridors. The purpose of this study is to develop a model which optimizes service patterns (SPs) and frequencies that yield minimum cost transit operation. Considering a general transit route with given stops and origin-destination demand, the proposed model consists of an objective total cost function and a set of constraints to ensure frequency conservation and sufficient capacity subject to operable fleet size. A numerical example is provided to demonstrate the effectiveness of the developed model, in which the demand and facility data of a rail transit route were given. Results show that the proposed model can be applied to optimize integrated SPs and headways that significantly reduce the total cost, while the resulting performance indicators are generated.     

Assessing the financial and social costs of public transport in differing operating environments and with endogenous demand

This paper uses a previously developed spreadsheet cost model which simulates public transport modes operated on a 12-km route to analyse the total costs of different passenger demand levels. The previous cost model was a very powerful tool to estimate the social and operator costs for different public transport technologies. However, as the model is strategic, some basic assumptions were made which are relaxed in this paper. First, the speed-flow equation in the original spreadsheet model assumes that speed decreases according to the ratio of the current frequency and the lane capacity which is based on the safety headway without taking into account passenger boardings. However, this may vary in different operating environments. Therefore, the speed-flow equation is improved by moving from a linear equation to a piecewise equation that considers the features of different operating environments. Second, the model assumes that supply is sufficient to meet demand. However, when the level of demand is high for the lower-capacity public transport technologies, passengers may find the incoming vehicle full and therefore, they have to wait more than one service interval. This paper applies queuing theory to investigate the probability of having to wait longer than the expected service headways which will affect the average passenger waiting time. The extra waiting time for each passenger is calculated and applied in the spreadsheet cost model. Third, the original model assumed that demand was externally fixed (exogenous). To evaluate the differences after applying these equations, endogenous demand rather than exogenous demand will be investigated by using the elasticities for passenger waiting time and journey time.     

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.     

Switching service types for multi-region bus systems

Conventional fixed-route bus services are generally preferred to flexible-route services at high demand densities, and vice versa. This paper formulates the problem of integrating conventional and flexible services that connect a main terminal to multiple local regions over multiple time periods. The system’s vehicle size, route spacing (for conventional services), service area (for flexible services), headways and fleet sizes are jointly optimized to minimize the sum of supplier costs and user costs. The route spacing for conventional bus services and service area for flexible bus services are also optimized for each region. The proposed solution method, which uses a genetic algorithm and analytic optimization, finds good solutions quickly. Numerical examples and sensitivity analyses confirm that the single fleet variable-type bus service may outperform either the single fleet conventional bus service or the single fleet flexible bus service when demand densities vary substantially among regions and time periods.     

Optimizing bus-size and headway in transit networks

Optimization models for calculating the best size for passenger carrying vehicles in urban areas were popular during the 1980s. These studies were abandoned in the ‘90s concluding that it was more efficient to use smaller buses at higher frequencies. This article returns to this controversial question, starting from the point of view that any calculation of bus size can only be made after considering the demand for each of the routes on the system. Therefore, an optimization model for sizing the buses and setting frequencies on each route in the system is proposed in accordance with the premises detailed below. The proposed model is a bi-level optimization model with constraints on bus capacity. The model allows buses of different sizes to be assigned to public transport routes optimizing the headways on each route in accordance with observed levels of demand. At the upper level the model considers the optimization of the system’s social and operating costs, these are understood to be the sum of the user’s and operator’s costs. At the lower level there is an assignment model for public transport with constraints on vehicle capacity which balances the flows for bus sizes and headways at each iteration. By graphically representing the results of the model applied to a real case, a series of useful conclusions are reached for the management and planning of a fleet of public transport vehicles.     

Investigating the irregularity of bus routes: highlighting how underlying assumptions of bus models impact the regularity results

A bus route is inherently unstable: when the system is uncontrolled, buses fail to maintain their time‐headways and tend to bunch. Several mathematical bus motion models were proposed to reproduce the bus behavior and assess management strategies. However, no work has established how the choice of a model impacts the irregularity of modeled bus systems, that is, the non‐respect of scheduled headways. Because of this gap, a large body of existing works assumes that the ability of these models to reproduce instability comes only from stochasticity, although the link between stochastic inputs and the level of irregularity remains unknown. Moreover, some recognized phenomena such as a change of travel conditions during a day or delays at signalized intersections are ignored. To address these shortcomings, this paper provides an overview of existing dynamic bus‐focused models and proposes a simple way to classify them. Commonly used deterministic and stochastic models are compared, which allows quantifying the relative influence of stochasticity of each model component on outputs. Moreover, we show that a change in the system equilibrium in a full deterministic system can lead to irregularity. Finally, this paper proposes a refinement of travel time models to account for non‐dynamic signals. In presence of traffic signals, we show that a bus system can be self‐regulated. Especially, these insights could help to calibrate bus model inputs to better reproduce real data. Copyright © 2014 John Wiley & Sons, Ltd.     

The marginal social cost of headway for a scheduled service

This brief paper derives the marginal social cost of headway for a scheduled service, i.e. the cost for users of marginal increases to the time interval between departures. In brief we may call it the value of headway in analogy with the value of travel time and the value of reliability. Users have waiting time costs as well as schedule delay costs measured relative to their desired time of arrival at the destination. They may either arrive at the station to choose just the next departure or they may plan for a specific departure in which case they incur also a planning cost. Then planning for a specific departure is costly but becomes more attractive at longer headways. Simple expressions for the user cost result. In particular, the marginal cost of headway is large at short headways and smaller at long headways. The difference in marginal costs is the value of time multiplied by half the headway.     

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.     

Analyzing transit service reliability using detailed data from automatic vehicular locator systems

The widespread adoption of automated vehicle location (AVL) systems and automatic passenger counters (APCs) in the transit industry has opened new venues in operations and system monitoring. In 2005, Metro Transit, Minnesota, implemented AVL system and partially implemented APC technologies. To date there has been little effort to employ the collected data in evaluating transit performance. This research uses such data to assess performance issues along a cross‐town route in the Metro Transit system. We generate a series of visual and analytical analyses to predict run time, schedule adherence and reliability of the transit route at two scales: the time point segment and the route level to demonstrate ways of identifying causes of decline in reliability levels. The analytical models show that while headways are maintained, schedule revisions are needed to improve run time and schedule adherence. Finally, the analysis suggests that many scheduled stops along this route are underutilized and recommends stop consolidation as a tool to decrease variability of service through concentrating passenger demand along a fewer number of stops. Copyright © 2010 John Wiley & Sons, Ltd.     

Integrating bus services with mixed fleets

Conventional bus service (with fixed routes and schedules) has lower average cost than flexible bus service (with demand-responsive routes) at high demand densities. At low demand densities flexible bus service has lower average costs and provides convenient door-to-door service. Bus size and operation type are related since larger buses have lower average cost per passenger at higher demand densities. The operation type and other decisions are jointly optimized here for a bus transit system connecting a major terminal to local regions. Conventional and flexible bus sizes, conventional bus route spacings, areas of service zones for flexible buses, headways, and fleet sizes are jointly optimized in multi-dimensional nonlinear mixed integer optimization problems. To solve them, we propose a hybrid approach, which combines analytic optimization with a Genetic Algorithm. Numerical analysis confirms that the proposed method provides near-optimal solutions and shows how the proposed Mixed Fleet Variable Type Bus Operation (MFV) can reduce total cost compared to alternative operations such as Single Fleet Conventional Bus (SFC), Single Fleet Flexible Bus (SFF), Mixed Fleet Conventional Bus (MFC) and Mixed Fleet Flexible Bus (MFF). With consistent system-wide bus sizes, capital costs are reduced by sharing fleets over times and over regions. The sensitivity of results to several important parameters is also explored.     

Distribution models for start-up lost time and effective departure flow rate

Due to its importance, lots of investigations had been carried out in the last four decades to study the relationship between phase duration and vehicle departure amount. In this paper, we aim to build appropriate distribution models for start-up lost time and effective departure flow rate, by considering their relations with the frequently mentioned departure headway distributions. The motivation behind is that distribution models could provide richer information than the conventional mean value models and thus better serve the need of traffic simulation and signal timing planning. To reach this goal, we first check empirical data collected in Beijing, China. Tests show that the departure headways at each position in a discharging queue are very weakly dependent or almost independent. Based on this new finding, two distribution models are proposed for start-up lost time and effective flow rate, respectively. We also examine the dependences of departure headways that are generated by three popular traffic simulation software: VISSIM, PARAMICS and TransModeler. Results suggest that in VISSIM, the departure headways at different positions are almost deterministically dependent and may not be in accordance with empirical observations. Finally, we discuss how the dependence of departure headways may influence traffic simulation and signal timing planning.     

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

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

Bus frequency determination using passenger count data

The importance of ridership information has led transit properties to increase the amount of manually collected data or alternatively to introduce automated surveillance techniques. Naturally, the bus operators are expected to gain useful information for operations planning by obtaining more accurate passenger counts. This paper describes and analyzes several appropriate data collection approaches for the bus operator in order to set the bus frequencies/headways efficiently. Four different methods are presented to derive the bus frequency: two are based on point check (maximum load) data and two propose the use of ride check (load profile) data. A ride check provides more complete information than a point check, but at a greater cost, and there is a question as to whether the additional information gained justifies the expense. Based on available old profiles, the four methods provide the bus scheduler with adequate guidance in selecting the type of data collection procedure. In addition, the scheduler can evaluate the minimum expected bus runs when the load standard is released and avoid overcrowding (in an average sense) at the same time. Alternative timetables are also investigated in conjunction with minimizing the required bus runs and number of buses for a single route. In this way, the derived headways can be analyzed within an acceptable range while considering the possible changes incurred indirectly to the fleet size. The integration between resource saving and frequency determination procedures allows the scheduler's performance to be improved.     

A stochastic transit assignment model using a dynamic schedule-based network

Using the schedule-based approach, in which scheduled time-tables 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 which consists of four components: in-vehicle time; waiting time; walking time; and a time penalty for each line change. With the exception of in-vehicle time, each of the other cost components is weighted by a sensitivity coefficient which varies among travelers and is defined by a density function. This time-dependent and stochastic minimum path is generated by a specially developed branch and bound algorithm. The assignment procedure is conducted over a period in which both passenger demand and train headways are varying. Due to the stochastic nature of the assignment problem, a Monte Carlo approach is employed to solve the problem. A case study using the Mass Transit Railway System in Hong Kong is given to demonstrate the model and its potential applications.