Planning hierarchical urban transit systems for reductions in greenhouse gas emissions

Public transit systems with high occupancy can reduce greenhouse gas (GHG) emissions relative to low-occupancy transportation modes, but current transit systems have not been designed to reduce environmental impacts. This motivates the study of the benefits of design and operational approaches for reducing the environmental impacts of transit systems. For example, transit agencies may replace level-of-service (LOS) by vehicle miles traveled (VMT) as a criterion in evaluating design and operational changes. In previous work, we explored the unintended consequences of lowering transit LOS on emissions in a single-technology transit system. Herein, we extend the analysis to account for a more realistic case: a transit system with a hierarchical structure (trunk and feeder lines) providing service to a city where demand is elastic. By considering the interactions between the trunk and the feeder systems, we provide a quantitative basis for designing and operating integrated urban transit systems that can reduce GHG emissions and societal costs. We find that highly elastic transit demand may cancel emission reduction potentials resulting from lowering LOS, due to demand shifts to lower occupancy vehicles. However, for mass transit modes, these potentials are still significant. Transit networks with buses, bus rapid transit or light rail as trunk modes should be designed and operated near the cost-optimal point when the demand is highly elastic, while this is not required for metro. We find that the potential for unintended consequences increases with the size of the city. Our results are robust to uncertainties in the costs and emissions parameters.     

Network equilibrium for congested multi‐mode networks with elastic demand

This paper proposes an elastic demand network equilibrium model for networks with transit and walking modes. In Hong Kong, the multi‐mode transit system services over 90% of the total journeys and the demand on it is continuously increasing. Transit and walking modes are related to each other as transit passengers have to walk to and from transit stops. In this paper, the multi‐mode elastic‐demand network equilibrium problem is formulated as a variational inequality problem where the combined mode and route choices are modeled in a hierarchical logit structures and the total travel demand for each origin‐destination pair is explicitly given by an elastic demand function. In addition, the capacity constraint for transit vehicles and the effects of bi‐directional flows on walkways are considered in the proposed model. All these congestion effects are taken into account for modeling the travel choices. A solution algorithm is developed to solve the multi‐mode elastic‐demand network equilibrium model. It is based on a Block Gauss‐Seidel decomposition approach coupled with the method of successive averages. A numerical example is used to illustrate the application of the proposed model and solution algorithm.     

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.     

Design of public transit network in urban area with elastic demand

This research focuses on an efficient design of transit network in urban areas. The system developed is used to create, analyze and optimize routes and frequencies of transit system in the network level. The analysis is based on elastic demand, so the shift of demand between modes in network due to different service level is of prime consideration. The developed system creates all feasible routes connecting all pairs of terminals in the network. Out of this vast pool of routes, a set of optimal routes is generated for a certain predetermined number that maintains connectivity of significant demand. Based on these generated routes, the system fulfils transportation demand by assigning demand that considers path and route choices for non-transit users and transit users. Together with the assignment of demand, transit frequencies are optimized and the related fleet-size is calculated. Having an optimal setting of solution, the system is continued by reconnecting the routes to find some other better solutions in the periphery of the optimal setting. A set of mathematical programming modules is developed. Real data from Sioux Falls city network is used to evaluate the performance of the model and compare with other heuristic methods.     

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.     

Evaluation of an existing bus network using a transit network optimisation model: a case study of the Hiroshima City Bus network

This study evaluates an existing bus network from the perspectives of passengers, operators, and overall system efficiency using the output of a previously developed transportation network optimisation model. This model is formulated as a bi-level optimisation problem with a transit assignment model as the lower problem. The upper problem is also formulated as bi-level optimisation problem to minimise costs for both passengers and operators, making it possible to evaluate the effects of reducing operator cost against passenger cost. A case study based on demand data for Hiroshima City confirms that the current bus network is close to the Pareto front, if the total costs to both passengers and operators are adopted as objective functions. However, the sensitivity analysis with regard to the OD pattern fluctuation indicates that passenger and operator costs in the current network are not always close to the Pareto front. Finally, the results suggests that, regardless of OD pattern fluctuation, reducing operator costs will increase passenger cost and increase inequity in service levels among passengers.     

Impact analysis of cordon-based congestion pricing on mode-split for a bimodal transportation network

This paper investigates the impact of cordon-based congestion pricing scheme on the mode-split of a bimodal transportation network with auto and rail travel modes. For any given toll-charge pattern, its impact on the mode-split can be estimated by solving a combined mode-split and traffic-assignment problem. Using a binary logit model for the mode-split, the combined problem is converted into a traffic-assignment problem with elastic demand. Probit-based stochastic user equilibrium (SUE) principle is adopted for this traffic-assignment problem, and a continuously distributed value of time (VOT) is assumed to convert the toll charges and transit fares into time-units. This combined mode-split and traffic-assignment problem is then formulated as a fixed-point model, which can be solved by a convergent Cost Averaging method. The combined mode-split and traffic-assignment problem is then used to analyze a multimodal toll design problem for cordon-based congestion pricing scheme, with the aim of increasing the mode-share of public transport system to a targeted level. Taking the fixed-point model as a constraint, the multimodal toll design problem is thus formulated as a mathematical programming with equilibrium constraints (MPEC) model. A genetic algorithm (GA) is employed to solve this MPEC model, which is then numerical validated by a network example.     

Transportation service procurement problem with transit time

In this work, we investigate transit time in transportation service procurement, which is conducted by shippers using auctions to purchase transportation service from carriers in the planning stage. Besides cost, we find that many shippers are most concerned with transit time in practice; shorter transit time indicates better transportation service. To minimize both the total cost and transit time, the problem faced by shippers is the biobjective transportation service procurement problem with transit time. To solve the problem, we introduce a biobjective integer programming model that can also accommodate some important business constraints. A biobjective branch-and-bound algorithm that finds all extreme supported nondominated solutions is developed. To speed up the algorithm, two fast feasibility checks, a network flow model for particular subproblems, and lower bounds from relaxation are proposed. In addition, a sophisticated heuristic is introduced to meet shipper’s requirements in some situations. Computational experiments on evaluating the performance of the algorithms are conducted on a set of test instances that are generated from practical data.     

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.     

Transit network design for small-medium size cities

This paper proposes a novel heuristic to solve the network design problem for public transport in small-medium size cities. Such cities can be defined as those with a diameter of a few kilometers with up to a few hundred thousand residents. These urban centers present a specific spatial configuration affecting the land use and mobility system. Transportation demand is widespread in origin and concentrated in a small number of attraction points close to each other. This particular structure of demand (‘many-to-few’) suggests the need for specific methodologies for the design of a transit system at a network level. In this paper, such design methodologies are defined in terms of models and solution procedures and tested on a selected case study. The solution methods show promising results. The key variables of the model are the routes and their frequencies. The constraints of the problem affect the overall demand to be served, the quality of the proposed service (transfer, load factors) and the definition of routes.     

On seat capacity in traffic assignment to a transit network

Seating or standing make distinct on‐board states to a transit rider, yielding distinct discomfort costs, with potential influence on the passenger route choice onto the transit network. The paper provides a transit assignment model that captures the seating capacity and its occupancy along any transit route. The main assumptions pertain to: the seat capacity by service route, selfish user behaviour, a seat allocation process with priority rules among the riders, according to their prior state either on‐board or at boarding. To each transit leg from access to egress station is associated a set of ‘service modes’, among which the riders are assigned in a probabilistic way, conditionally on their priority status and the ratio between the available capacity and the flow of them. Thus the leg cost is a random variable, with mean value to be included in the trip disutility. Computationally efficient algorithms are provided for, respectively, loading the leg flows and evaluating the leg costs along a transit line. At the network level, a hyperpath formulation is provided for supply‐demand equilibrium, together with a property of existence and an method of successive averages equilibration algorithm. It is shown that multiple equilibria may arise. Copyright © 2010 John Wiley & Sons, Ltd.     

Transit network design with allocation of green vehicles: A genetic algorithm approach

The use of fossil fuels in transportation generates harmful emissions that accounts for nearly half of the total pollutants in urban areas. Dealing with this issue, local authorities are dedicating specific efforts to seize the opportunity offered by new fuels and technological innovations in achieving a cleaner urban mobility. In fact, authorities are improving environmental performances of their public transport fleet by procuring cleaner vehicles, usually called low and zero emission vehicles (LEV and ZEV, respectively). Nevertheless there seems to be a lack of methodologies for supporting stakeholders in decisions related to the introduction of green vehicles, whose allocation should be performed since the network design process in order to optimize their available green capacity.In this paper, the problem of clean vehicle allocation in an existing public fleet is faced by introducing a method for solving the transit network design problem in a multimodal, demand elastic urban context dealing with the impacts deriving from transportation emissions.The solving procedure consists of a set of heuristics which includes a routine for route generation and a genetic algorithm for finding a sub-optimal set of routes with the associated frequencies.     

Transit network design by genetic algorithm with elitism

The transit network design problem is concerned with the finding of a set of routes with corresponding schedules for a public transport system. This problem belongs to the class of NP-Hard problem because of the vast search space and multiple constraints whose optimal solution is really difficult to find out. The paper develops a Population based model for the transit network design problem. While designing the transit network, we give preference to maximize the number of satisfied passengers, to minimize the total number of transfers, and to minimize the total travel time of all served passengers. Our approach to the transit network design problem is based on the Genetic Algorithm (GA) optimization. The Genetic Algorithm is similar to evolution strategy which iterates through fitness assessment, selection and breeding, and population reassembly. In this paper, we will show two different experimental results performed on known benchmark problems. We clearly show that results obtained by Genetic Algorithm with increasing population is better than so far best technique which is really difficult for future researchers to beat.     

Robust transit network design with stochastic demand considering development density

This paper analyzes the influence of urban development density on transit network design with stochastic demand by considering two types of services, rapid transit services, such as rail, and flexible services, such as dial-a-ride shuttles. Rapid transit services operate on fixed routes and dedicated lanes, and with fixed schedules, whereas dial-a-ride services can make use of the existing road network, hence are much more economical to implement. It is obvious that the urban development densities to financially sustain these two service types are different. This study integrates these two service networks into one multi-modal network and then determines the optimal combination of these two service types under user equilibrium (UE) flows for a given urban density. Then we investigate the minimum or critical urban density required to financially sustain the rapid transit line(s). The approach of robust optimization is used to address the stochastic demands as captured in a polyhedral uncertainty set, which is then reformulated by its dual problem and incorporated accordingly. The UE principle is represented by a set of variational inequality (VI) constraints. Eventually, the whole problem is linearized and formulated as a mixed-integer linear program. A cutting constraint algorithm is adopted to address the computational difficulty arising from the VI constraints. The paper studies the implications of three different population distribution patterns, two CBD locations, and produces the resultant sequences of adding more rapid transit services as the population density increases.     

The design of single allocation incomplete hub networks

The hub location problem deals with finding the location of hub facilities and allocating the demand nodes to these hub facilities so as to effectively route the demand between any origin–destination pair. In the extensive literature on this challenging network design problem, it has widely been assumed that the subgraph induced by the hub nodes is complete. Relaxation of this basic assumption constitutes the starting point of the present work. In this study, we provide a uniform modeling treatment to all the single allocation variants of the existing hub location problems, under the incomplete hub network design. No network structure other than connectivity is imposed on the induced hub network. Within this context, the single allocation incomplete p-hub median, the incomplete hub location with fixed costs, the incomplete hub covering, and the incomplete p-hub center network design problems are defined, and efficient mathematical formulations for these problems with O(n³) variables are introduced. Computational analyses with these formulations are presented on the various instances of the CAB data set and on the Turkish network.     

Equity-oriented skip-stopping schedule optimization in an oversaturated urban rail transit network

In this study, we focus on improving system-wide equity performance in an oversaturated urban rail transit network based on multi-commodity flow formulation. From the system perspective, an urban rail transit network is a distributed system, where a set of resources (i.e., train capacity) is shared by a number of users (i.e., passengers), and equitable individuals and groups should receive equal shares of resources. However, when oversaturation occurs in an urban rail transit network during peak hours, passengers waiting at different stations may receive varying shares of train capacity leading to the inequity problem under train all-stopping pattern. Train skip-stopping pattern is an effective operational approach, which holds back some passengers at stations and re-routes their journeys in the time dimension based on the available capacity of each train. In this study, the inequity problem in an oversaturated urban rail transit network is analyzed using a multi-commodity flow modeling framework. In detail, first, discretized states, corresponding to the number of missed trains for passengers, are constructed in a space-time-state three-dimensional network, so that the system-wide equity performance can be viewed as a distribution of all passengers in different states. Different from existing flow-based optimization models, we formulate individual passenger and train stopping pattern as commodity and network structure in the multi-commodity flow-modeling framework, respectively. Then, we aim to find an optimal commodity flow and well-designed network structure through the proposed multi-commodity flow model and simultaneously achieve the equitable distribution of all passengers and the optimal train skip-stopping pattern. To quickly solve the proposed model and find an optimal train skip-stopping pattern with preferable system-wide equity performance, the proposed linear programming model can be effectively decomposed to a least-cost sub-problem with positive arc costs for each individual passenger and a least-cost sub-problem with negative arc costs for each individual train under a Lagrangian relaxation framework. For application and implementation, the proposed train skip-stopping optimization model is applied to a simple case and a real-world case based on Batong Line in the Beijing Subway Network. The simple case demonstrates that our proposed Lagrangian relaxation framework can obtain the approximate optimal solution with a small-gap lower bound and a lot of computing time saved compared with CPLEX solver. The real-world case based on Batong Line in the Beijing Subway Network compares the equity and efficiency indices under the operational approach of train skip-stopping pattern with those under the train all-stopping pattern to state the advantage of the train skip-stopping operational approach.     

A comparison of user-optimum versus system-optimum traffic assignment in transportation network design

In this report, we compare the computational efficiency and results of solving two alternative models for the problem of determining improvements to an urban road network. Using a 1462 link, 584 node test network of the north Dallas area, we compare a model which assumes user-optimum behavior of travelers with a model which assumes system-optimum flows. Both of these models allow improvements to the road network to take on any nonnegative value, rather than requiring discrete improvement values. Investment costs are modeled by functions with decreasing marginal costs. Unfortunately, the user-optimum model, which is much more realistic than the system-optimum one, normally cannot be solved optimally. However, the simpler system-optimum model can be optimally solved, provided that investment costs are approximated by linear functions. Thus, for this network design problem we compare an accurate representation which can be solved only approximately with an approximate representation which can be solved optimally. Our computational testing showed that the system-optimum model produces solutions as good as those from the user-optimum model, and thus seems justified when favored by other considerations, such as ease of coding, availability of “canned” programs, etc.     

An integrated measure of accessibility and reliability of mass transit systems

The growth of a city or a metropolis requires well-functioning transit systems to accommodate the ensuing increase in travel demand. As a result, mass transit networks have to develop and expand from simple to complex topological systems over time to meet this demand. Such an evolution in the networks’ structure entails not only a change in network accessibility, but also a change in the level of network reliability on the part of stations and the entire system as well. Network accessibility and reliability are popular measures that have been widely applied to evaluate the resilience and vulnerability of a spatially networked system. However, the use of a single measure, either accessibility or reliability, provides different results, which demand an integrated measure to evaluate the network’s performance comprehensively. In this paper, we propose a set of integrated measures, named ACCREL (Integrated Accessibility and Reliability indicators) that considers both metrics in combination to evaluate a network’s performance and vulnerability. We apply the new measures for hypothetical mass transit system topologies, and a case study of the metro transit system in Seoul follows, highlighting the dynamics of network performance with four evolutionary stages. The main contribution of this study lies in the results from the experiments, which can be used to inform how transport network planning can be prepared to enhance the network functionality, thereby achieving a well-balanced, accessible, and reliable system. Insights on network vulnerability are also drawn for public transportation planners and spatial decision makers.     

Network design for personal rapid transit under transit-oriented development

This study addresses guideway network design for personal rapid transit (PRT) favoring transit-oriented development. The guideway network design problem seeks to minimize both the guideway construction cost and users’ travel time. In particular, a set of optional points, known as Steiner points, are introduced in the graph to reduce the guideway length. The model is formulated as a combined Steiner and assignment problem, and a Lagrangian relaxation based solution algorithm is developed to solve the optimal solution. Numerical studies are carried on a real-sized network, and illustrate that the proposed model and solution algorithm can solve the PRT guideway network design problem effectively.     

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

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