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.     

Ridership estimation procedure for a transit corridor with new bus rapid transit service

Ridership estimation is a critical step in the planning of a new transit route or change in service. Very often, when a new transit route is introduced, the existing routes will be modified, vehicle capacities changed, or service headways adjusted. This has made ridership forecasts for the new, existing, and modified routes challenging. This paper proposes and demonstrates a procedure that forecasts the ridership of all transit routes along a corridor when a new bus rapid transit (BRT) service is introduced and existing regular bus services are adjusted. The procedure uses demographic data along the corridor, a recent origin–destination survey data, and new and existing transit service features as inputs. It consists of two stages of transit assignment. In the first stage, a transit assignment is performed with the existing transit demand on the proposed BRT and existing bus routes, so that adjustments to the existing bus services can be identified. This transit assignment is performed iteratively until there is no adjustment in transit services. In the second stage, the transit assignment is carried out with the new BRT and adjusted regular bus services, but incorporates a potential growth in ridership because of the new BRT service. The final outputs of the procedure are ridership for all routes and route segments, boarding and alighting volumes at all stops, and a stop‐by‐stop trip matrix. The proposed ridership estimation procedure is applicable to a new BRT route with and without competing regular bus routes and with BRT vehicles traveling in dedicated lanes or in mixed traffic. The application of the proposed procedure is demonstrated via a case study along the Alameda Corridor in El Paso, Texas. Copyright © 2015 John Wiley & Sons, Ltd.     

Demand responsive transit systems with time-dependent demand: User equilibrium,system optimum,and management strategy

The operating cost of a demand responsive transit (DRT) system strictly depends on the quality of service that it offers to its users. An operating agency seeks to minimize operating costs while maintaining the quality of service while users experience costs associated with scheduling, waiting, and traveling within the system. In this paper, an analytical model is employed to approximate the agency's operating cost for running a DRT system with dynamic demand and the total generalized cost that users experience as a result of the operating decisions. The approach makes use of Vickrey's (1969) congestion theory to model the dynamics of the DRT system in the equilibrium condition and approximate the generalized cost for users when the operating capacity is inadequate to serve the time-dependent demand over the peak period without excess delay. The efficiency of the DRT system can be improved by optimizing one of three parameters that define the agency's operating decision: (1) the operating capacity of the system, (2) the number of passengers that have requested a pick-up and are awaiting service, and (3) the distribution of requested times for service from the DRT system. A schedule management strategy and dynamic pricing strategies are presented that can be implemented to manage demand and reduce the total cost of the DRT system by keeping the number of waiting requests optimized over the peak period. In the end, proposed optimization strategies are compared using a numerical example.     

Bus rapid transit systems: a comparative assessment

There is renewed interest in many developing and developed countries in finding ways of providing efficient and effective public transport that does not come with a high price tag. An increasing number of nations are asking the question—what type of public transport system can deliver value for money? Although light rail has often been promoted as a popular ‘solution’, there has been progressively emerging an attractive alternative in the form of bus rapid transit (BRT). BRT is a system operating on its own right-of-way either as a full BRT with high quality interchanges, integrated smart card fare payment and efficient throughput of passengers alighting and boarding at bus stations; or as a system with some amount of dedicated right-of-way (light BRT) and lesser integration of service and fares. The notion that buses essentially operate in a constrained service environment under a mixed traffic regime and that trains have privileged dedicated right-of-way, is no longer the only sustainable and valid proposition. This paper evaluates the status of 44 BRT systems in operation throughout the world as a way of identifying the capability of moving substantial numbers of passengers, using infrastructure whose costs overall and per kilometre are extremely attractive. When ongoing lifecycle costs (operations and maintenance) are taken into account, the costs of providing high capacity integrated BRT systems are an attractive option in many contexts.     

Optimization of bus stop locations for improving transit accessibility

A mathematical model is developed in this paper to improve the accessibility of a bus service. To formulate the optimization model, a segment of a bus route is given, on which a number of demand entry points are distributed realistically. The objective total cost function (i.e. the sum of supplier and user costs) is minimized by optimizing the number and locations of stops, subject to non‐additive users' value of time. A numerical example is designed to demonstrate the effectiveness of the method thus developed to optimize the bus stop location problem. The sensitivity of the total cost to various parameters (e.g. value of users' time, access speed, and demand density) and the effect of the parameters on the optimal stop locations are analyzed and discussed.     

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.     

Optimal design of a short‐turning strategy considering seat availability

We develop a methodology to optimize the schedule coordination of a full‐stop service pattern and a short‐turning service pattern on a bus route. To capture the influence of bus crowding and seat availability on passengers' riding experience, we develop a Markov model to describe the seat‐searching process of a passenger and an approach to estimate the transition probabilities of the Markov model. An optimization model that incorporates the Markov model is proposed to design the short‐turning strategy. The proposed model minimizes the total cost, which includes operational cost, passengers' waiting time cost and passengers' in‐vehicle travel time cost. Algorithm is developed to produce optimal values of the decision variables. The proposed methodology is evaluated in a case study. Compared with methodologies that ignore the effect of bus crowding, the proposed methodology could better balance bus load along the route and between two service patterns, provide passengers with better riding experience and reduce the total cost. In addition, it is shown that the optimal design of the short‐turning strategy is sensitive to seat capacity. Copyright © 2016 John Wiley & Sons, Ltd.     

Investigating the impact of stochastic vehicle arrivals to optimal stop spacing and headway for a feeder bus route

Stop spacing and service frequency (i.e., the inverse of headway) are key elements in transit service planning. The trade‐offs between increasing accessibility and reducing travel time, which affect transit system performance, need to be carefully evaluated. The objective of this study is to optimize stop spacing and headway for a feeder bus route, considering the relationship between the variance of inter‐arrival time (VIAT), which yields the minimum total cost (including user and operator costs). A solution algorithm, called successive substitution, is adapted to efficiently search for the optimal solutions. In a numerical example, the developed model is applied to planning a feeder bus route in Newark, New Jersey. The results indicate that the optimal stop spacing should be longer that those suggested by previous studies where the impact of VIAT was ignored. Reducing VIAT via certain operational control strategies (i.e., holding/stop‐skipping, transit signal priority) may shorten stop spacing and improve accessibility. Copyright © 2014 John Wiley & Sons, Ltd.     

Demand-driven timetable design for metro services

Timetable design is crucial to the metro service reliability. A straightforward and commonly adopted strategy in daily operation is a peak/off-peak-based schedule. However, such a strategy may fail to meet dynamic temporal passenger demand, resulting in long passenger waiting time at platforms and over-crowding in trains. Thanks to the emergence of smart card-based automated fare collection systems, we can now better quantify spatial–temporal demand on a microscopic level. In this paper, we formulate three optimization models to design demand-sensitive timetables by demonstrating train operation using equivalent time (interval). The first model aims at making the timetable more dynamic; the second model is an extension allowing for capacity constraints. The third model aims at designing a capacitated demand-sensitive peak/off-peak timetable. We assessed the performance of these three models and conducted sensitivity analyzes on different parameters on a metro line in Singapore, finding that dynamical timetable built with capacity constraints is most advantageous. Finally, we conclude our study and discuss the implications of the three models: the capacitated model provides a timetable which shows best performance under fixed capacity constraints, while the uncapacitated model may offer optimal temporal train configuration. Although we imposed capacity constraints when designing the optimal peak/off-peak timetable, its performance is not as good as models with dynamical headways. However, it shows advantages such as being easier to operate and more understandable to the passengers.     

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

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

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.     

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.     

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.     

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.     

快速公交系统站台停靠时间模型研究

快速公交系统停靠站台停车延误是影响快速公交运行车速的关键因素之一,因此构建快速公交系统站台停靠时间模型是提升快速公交服务水平的基础理论研究。本文选取盐城BRT-1号线的起始站、中途站、客流离散站等三类站点为研究对象,综合运用数理统计法与数据挖掘法,构建快速公交系统站台停靠时间模型,并对该模型的合理性进行了检验。研究表明:盐城市BRT-1号线三类站台的快速公交车辆停靠时间与上下车乘客人数呈线性关系,即快速公交车辆停靠时间与上下车乘客人数的检验参数R2均大于0.8。     

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.     

A multiobjective planning model for intercity train seat allocation

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

Continuous approximation for skip-stop operation in rail transit

Operating speed of a transit corridor is a key characteristic and has many consequences on its performance. It is generally accepted that an increased operating speed for a given fleet leads to reduced operating costs (per kilometer), travel and waiting times (three changes that can be computed precisely), an improved comfort and level of service, which can attract new passengers who are diverted from automobile (items harder to estimate precisely). That is why several operation schemes which aim to increase the operating speed are studied in the literature, such as deadheading, express services, and stop skipping.A novel category of solutions to this problem for one-way single-track rail transit is to perform accelerated transit operations with fixed stopping schedules. The concept is quite simple: as the time required for stopping at each station is an important part of travel time, reducing it would be a great achievement. Particular operations that take advantage of this idea already exist. This paper focuses on one of them: the skip-stop operation for rail transit lines using a single one-way track. It consists in defining three types of stations: AB stations where all the trains stop, and A and B stations where only half of the trains stop (stations type A and B are allocated interchangeably). This mode of operation is already described in the literature (Vuchic, 1973, Vuchic, 1976, Vuchic, 2005) and has been successfully implemented in the Metro system of Santiago, Chile.This work tackles the problem with a continuous approximation approach. The problem is described with a set of geographically dependent continuous parameters like the density of stations for a given line. Cost functions are built for a traditional (all-stop) operation and for skip-stop operation as described above. A simple example is presented to support this discussion. Finally, a discussion about the type of scenarios in which skip-stop operations are more beneficial is presented.     

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.     

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.