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.     

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.     

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.     

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.     

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.     

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 structured flexible transit system for low demand areas

Public transit structure is traditionally designed to contain fixed bus routes and predetermined bus stations. This paper presents an alternative flexible-route transit system, in which each bus is allowed to travel across a predetermined area to serve passengers, while these bus service areas collectively form a hybrid “grand” structure that resembles hub-and-spoke and grid networks. We analyze the agency and user cost components of this proposed system in idealized square cities and seek the optimum network layout, service area of each bus, and bus headway, to minimize the total system cost. We compare the performance of the proposed transit system with those of comparable systems (e.g., fixed-route transit network and taxi service), and show how each system is advantageous under certain passenger demand levels. It is found out that under low-to-moderate demand levels, the proposed flexible-route system tends to have the lowest system cost.     

A connected-vehicle-based dynamic control model for managing the bus bunching problem with capacity constraints

This paper describes a connected-vehicle-based system architecture which can provide more precise and comprehensive information on bus movements and passenger status. Then a dynamic control method is proposed using connected vehicle data. Traditionally, the bus bunching problem has been formulated into one of two types of optimization problem. The first uses total passenger time cost as the objective function and capacity, safe headway, and other factors as constraints. Due to the large number of scenarios considered, this type of framework is inefficient for real-time implementation. The other type uses headway adherence as the objective and applies a feedback control framework to minimize headway variations. Due to the simplicity in the formulation and solution algorithms, the headway-based models are more suitable for real-time transit operations. However, the headway-based feedback control framework proposed in the literature still assumes homogeneous conditions at all bus stations, and does not consider restricting passenger loads within the capacity constraints. In this paper, a dynamic control framework is proposed to improve not only headway adherence but also maintain the stability of passenger load within bus capacity in both homogenous and heterogeneous situations at bus stations. The study provides the stability conditions for optimal control with heterogeneous bus conditions and derives optimal control strategies to minimize passenger transit cost while maintaining vehicle loading within capacity constraints. The proposed model is validated with a numerical analysis and case study based on field data collected in Chengdu, China. The results show that the proposed model performs well on high-demand bus routes.     

Optimally combined headway and timetable reliable public transport system

This paper presents a model-based multiobjective control strategy to reduce bus bunching and hence improve public transport reliability. Our goal is twofold. First, we define a proper model, consisting of multiple static and dynamic components. Bus-following model captures the longitudinal dynamics taking into account the interaction with the surrounding traffic. Furthermore, bus stop operations are modeled to estimate dwell time. Second, a shrinking horizon model predictive controller (MPC) is proposed for solving bus bunching problems. The model is able to predict short time-space behavior of public transport buses enabling constrained, finite horizon, optimal control solution to ensure homogeneity of service both in time and space. In this line, the goal with the selected rolling horizon control scheme is to choose a proper velocity profile for the public transport bus such that it keeps both timetable schedule and a desired headway from the bus in front of it (leading bus). The control strategy predicts the arrival time at a bus stop using a passenger arrival and dwell time model. In this vein, the receding horizon model predictive controller calculates an optimal velocity profile based on its current position and desired arrival time. Four different weighting strategies are proposed to test (i) timetable only, (ii) headway only, (iii) balanced timetable - headway tracking and (iv) adaptive control with varying weights. The controller is tested in a high fidelity traffic simulator with realistic scenarios. The behavior of the system is analyzed by considering extreme disturbances. Finally, the existence of a Pareto front between these two objectives is also demonstrated.     

Optimal bus service patterns and frequencies considering transfer demand elasticity with genetic algorithm

This paper attempts to optimize bus service patterns (i.e., all-stop, short-turn, and express) and frequencies which minimize total cost, considering transfer demand elasticity. A mathematical model is developed based on the objective total cost for a generalized bus route, which is optimized subject to a set of constraints ensuring sufficient capacity, an operable bus fleet, and service frequency conservation. To optimize the integrated service of a bus route with many stops, which is a combinatorial optimization problem, a genetic algorithm is developed and applied to search for the solution. A case study, based on a real-world bus route in New Jersey, is conducted to demonstrate the applicability and effectiveness of the developed model and the solution algorithm. Results show that the proposed methodology is fairly efficient, and the optimized bus service significantly reduces total cost.     

Optimization Of Headway,Vehicle Size and Route Choice for Minimum Cost Feeder Service

ABSTRACT

Many people use public transportation systems to reach their destination, while others use personal vehicles. Poor transportation systems do not attract ridership. Therefore, the usage of passenger cars increases, and traffic and environmental conditions deteriorate. Efficient public transportation has been recognized as one of the potential ways of mitigating air pollution, reducing energy consumption, improving mobility and alleviating traffic congestion. The objective of this study is to optimize a bus feeder service that provides the shuttle service between a recreation center (e.g. Sandy Hook, NJ) and a major public transportation facility, subject to site-specific constraints such as vehicle schedules, bus availability, service capacity and budget. The decision variables include bus headway, vehicle size and route choice. The solution methodology integrating both analytical and numerical techniques is developed, which optimizes the decision variables. Finally, the proposed solution methodology is applied to a case study. Numerical results, including optimal solutions and sensitivity analyses, are presented while the level of coordination between the feeder service and a major transportation service is discussed.     

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.     

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.     

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.     

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.     

Optimizing fare and headway to facilitate timed transfer considering demand elasticity

Fare and service frequency significantly affect transit users’ willingness to ride, as well as the supplier's revenue and operating costs. To stimulate demand and increase productivity, it is desirable to reduce the transfer time from one route to another via efficient service coordination, such as timed transfer. Since demand varies both temporally and spatially, it may not be cost-effective to synchronize vehicle arrivals on all connecting routes at a terminal. In this paper, we develop a schedule coordination model to optimize fare and headway considering demand elasticity. The headway of each route is treated as an integer-multiple of a base common headway. A discounted (reduced) fare is applied as an incentive to encourage ridership and, thus, stimulate public transit usage. The objective of the proposed coordination model is used to maximize the total profit subject to the service constraint. A numerical example is given to demonstrate the applicability of the proposed model. The results show that the optimized fare and headway may be carefully applied to yield the maximum profit. The relationship between the decision variables and model parameters is explored in the sensitivity analysis.     

Optimizing train operational plan in an urban rail corridor based on the maximum headway function

The train operational plan (TOP) plays a crucial role in the efficient and effective operation of an urban rail system. We optimize the train operational plan in a special network layout, an urban rail corridor with one terminal yard, by decomposing it into two sub-problems, i.e., the train departure profile optimization and the rolling stock circulation optimization. The first sub-problem synthetically optimizes frequency setting, timetabling and the rolling stock circulation at the terminal without a yard. The maximum headway function is generated to ensure the service of the train operational plan without considering travel demand, then we present a model to minimize the number of train trips, and design a heuristic algorithm to maximize the train headway. On the basis of a given timetable, the rolling stock circulation optimization only involves the terminal with a yard. We propose a model to minimize the number of trains and yard–station runs, and an algorithm to find the optimal assignment of train-trip pair connections is designed. The computational complexities of the two algorithms are both linear. Finally, a real case study shows that the train operational plan developed by our approach enables a better match of train headway and travel demand, and reduces the operational cost while satisfying the requirement of the level of service.     

An analytic investigation of optimal bus size

The use of smaller buses offers passengers a better service frequency for a given service capacity, but costs more to operate per seat provided. Within this trade-off there is an optimal bus size which maximises social benefit. A mathematical model is described which can be solved analytically to provide an explicit relationship between optimal bus size and factors such as operating cost, level of demand, and demand elasticities. The model includes: passenger demand varying with the generalised cost of travel according to a constant elasticity; the effect of changes in bus occupancy on average waiting times and on operating speed; the financial constraint that farebox revenue must equal operating cost less subsidy; an allowance for external benefits associated with generated demand, and for the effect of the flow of buses on traffic congestion; and an operating cost increasing linearly with bus size. The optimal size varies with the square root of demand, and with the unit cost to the power of 0.1 to 0.2. It also increases slowly with the proportion of cost covered by subsidy. For typical urban operating conditions in the United Kingdom the optimal size for a monopoly service lies between 55 and 65 seats assuming the observed relationship between cost and size; it is possible that changes in working practices could make smaller buses relatively cheaper to operate, so reducing the optimal size, but it seems unlikely to fall below 40 seats.     

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.     

Designing large-scale bus network with seasonal variations of demand

Creating a bus network that covers passenger demand conveniently is an important ingredient of the transit operations planning process. Certainly determination of optimal bus network is highly sensitive to any change of demand, thus it is desirable not to consider average or estimated figures, but to take into account prudently the variations of the demand. Many cities worldwide experience seasonal demand variations which naturally have impact on the convenience and optimality of the transit service. That is, the bus network should provide convenient service across all seasons. This issue, addressed in this work, has not been thoroughly dealt with neither in practice nor in the literature. Analyzing seasonal transit demand variations increases further the computational complexity of the bus-network design problem which is known as a NP-hard problem. A solution procedure using genetic algorithm efficiently, with a defined objective-function to attain the optimization, is proposed to solve this cumbersome problem. The method developed is applied to two benchmarked networks and to a case study, to the city of Mashhad in Iran with over 3.2 million residents and 20 million visitors annually. The case study, characterized by a significant seasonal demand variation, demonstrates how to find the best single network of bus routes to suit the fluctuations of the annual passenger demand. The results of comparing the proposed algorithm to previously developed algorithms show that the new development outperforms the other methods between 1% and 9% in terms of the objective function values.