Optimal feeder bus routes on irregular street networks

The methodology presented here seeks to optimize bus routes feeding a major intermodal transit transfer station while considering intersection delays and realistic street networks. A model is developed for finding the optimal bus route location and its operating headway in a heterogeneous service area. The criterion for optimality is the minimum total cost, including supplier and user costs. Irregular and discrete demand distributions, which realistically represent geographic variations in demand, are considered in the proposed model. The optimal headway is derived analytically for an irregularly shaped service area without demand elasticity, with non‐uniformly distributed demand density, and with a many‐to‐one travel pattern. Computer programs are designed to analyze numerical examples, which show that the combinatory type routing problem can be globally optimized. The improved computational efficiency of the near‐optimal algorithm is demonstrated through numerical comparisons to an optimal solution obtained by the exhaustive search (ES) algorithm. The CPU time spent by each algorithm is also compared to demonstrate that the near‐optimal algorithm converges to an acceptable solution significantly faster than the ES algorithm.     

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.     

A model for estimating the optimal cycle length of demand responsive feeder transit services

The general lack of first/last mile connectivity is one of the main challenges faced by today’s public transit. One of the possible actions towards a solution to this problem is the planning, design and implementation of efficient feeder transit services. This paper develops an analytical model which allows for an easy computation of near optimal terminal-to-terminal cycle length of a demand responsive feeder service to maximize service quality provided to customers, defined as the inverse of a weighted sum of waiting and riding times. The model estimates the recommended cycle length by only plugging in geometrical parameters and demand data, without relying on extensive simulation analyses or rule of thumbs. Simulation experiments and comparisons with real services validate our model, which would allow planners, decision makers and practitioners to quickly identify the best feeder transit operating design of any given residential area.     

Intermodal Transit System Coordination

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

Optimization of bus routing strategies for evacuation

Efficient transportation of evacuees during an emergency has long been recognized as a challenging issue. This paper investigates emergency evacuation strategies that rely on public transit, where buses run continuously, rather than fixed route, based upon the spatial and temporal information of evacuee needs. We formulated an optimal bus operating strategy that minimizes the exposed casualty time rather than operational cost, as a deterministic mixed‐integer program, and investigated the solution algorithm. A Lagrangian‐relaxation‐based solution algorithm was developed for the proposed model. Numerical experiments with different problem sizes were conducted to evaluate the method. Copyright © 2013 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.     

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.     

Containership scheduling with transit-time-sensitive container shipment demand

This paper examines the optimal containership schedule with transit-time-sensitive demand that is assumed to be a decreasing continuous function of transit time. A mixed-integer nonlinear non-convex optimization model is first formulated to maximize the total profit of a ship route. In view of the problem structure, a branch-and-bound based holistic solution method is developed. It is rigorously demonstrated that this solution method can obtain an ε-optimal solution in a finite number of iterations for general forms of transit-time-sensitive demand. Computational results based on a trans-Pacific liner ship route demonstrate the applicability and efficiency of the solution method.     

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.     

Modeling transit technology selection in a linear transportation corridor

This paper proposes an analytical model for investigating transit technology selection problem from a perspective of transit authority. Given a transit technology alternative (e.g., metro, light rail transit, or bus rapid transit), the proposed model aims to maximize the social welfare of the transit system by determining the optimal combination of transit line length, number of stations, station location (or spacing), headway, and fare. In the proposed model, the effects of passenger demand elasticity and capacity constraint are explicitly considered. The properties of the model are examined analytically, and a heuristic solution procedure for determining the model solution is presented. By comparing the optimized social welfare for different transit technology alternatives, the optimal transit technology solution can be obtained together with critical population density. On the basis of a simple population growth rate formula, optimal investment timing of a new transit technology can be estimated. The proposed methodology is illustrated in several Chinese cities. Insightful findings are reported on the interrelation among transit technology selection, population density, transit investment cost, and transit line parameter design as well as the comparison between social welfare maximization and profit maximization regimes. Copyright © 2014 John Wiley & Sons, Ltd.     

Stochastic dynamic switching in fixed and flexible transit services as market entry-exit real options

The first analytical stochastic and dynamic model for optimizing transit service switching is proposed for “smart transit” applications and for operating shared autonomous transit fleets. The model assumes a region that requires many-to-one last mile transit service either with fixed-route buses or flexible-route, on-demand buses. The demand density evolves continuously over time as an Ornstein-Uhlenbeck process. The optimal policy is determined by solving the switching problem as a market entry and exit real options model. Analysis using the model on a benchmark computational example illustrates the presence of a hysteresis effect, an indifference band that is sensitive to transportation system state and demand parameters, as well as the presence of switching thresholds that exhibit asymmetric sensitivities to transportation system conditions. The proposed policy is computationally compared in a 24-hour simulation to a “perfect information” set of decisions and a myopic policy that has been dominant in the flexible transit literature, with results that suggest the proposed policy can reduce by up to 72% of the excess cost in the myopic policy. Computational experiments of the “modular vehicle” policy demonstrate the existence of an option premium for having flexibility to switch between two vehicle sizes.     

Fixed‐route taxi system: route network design and fleet size minimization problems

This paper introduces the taxi route network design problem (TXRNDP) for a fixed‐route taxi service operating in Iran and, in similar form, in various other developing countries. The service operates fairly similar to regular transit services in that vehicles are only permitted to follow a certain predetermined route on the network. The service is provided with small size vehicles and main features are that vehicles only depart if full and that there are no intermediate boarding stops. In Iran the service attracts a high modal share but requires better coordination which is the main motivation for the present study. We develop a mathematical programming model to minimize the total travel time experienced by passengers while constraining the number of taxi lines, the trip transfer ratio and the length of taxi lines. A number of assumptions are introduced in order to allow finding an exact rather than heuristic solution. We further develop a linear programming solution to minimize the number of taxis required to serve the previously found fixed‐route taxi network. Results of a case study with the city of Zanjan, Iran, illustrate the resulting taxi flows and suggest the capability of the proposed model to reduce the total travel time, the total waiting time and the number of taxi lines compared to the current taxi operation. Copyright © 2016 John Wiley & Sons, Ltd.     

Integrated smart feeder/shuttle transit service: simulation of new routing strategies

The idea of designing an integrated smart feeder/shuttle service stemmed from the need to overcome the problem of using an excessive number of cars arriving and parking at a train station within the same time span. This problem results in high parking demand around the train station. Moreover, some potential train riders will, instead, use their cars and hence become a party to increasing the traffic congestion. This work develops a new idea of an integrated and innovative feeder/shuttle system with new operating and routing concepts. The fulfilled objectives are as follows: (i) to construct and examine different operating strategies from both the user and operator perspectives; (ii) to examine different routing models and scenarios; and (iii) to construct a simulation tool for (i) and (ii). Ten different routing strategies are examined, with all the combinations of fixed/flexible routes, fixed/flexible schedules, a unidirectional or bidirectional concept, and shortcut (shortest path) and/or short‐turn (turnaround) concepts. These strategies are investigated by employing a simulation model specifically developed and constructed for this purpose. This simulation model is used in a case study of Castro Valley in California in which the feeder/shuttle service is coordinated with the Bay Area Rapid Transit service, and the 10 routing strategies are compared in regard to four fleet‐size scenarios. One of the interesting results found is that the fixed‐route and flexible‐route concepts are comparable in performance measures when applying a combination of operating strategies. Copyright © 2011 John Wiley & Sons, Ltd.     

A methodology for choosing between fixed‐route and flex‐route policies for transit services

Flex‐route transit brings together the low cost operability of fixed‐route transit with the flexibility of demand responsive transit, and in recent years, it has become the most popular type of flexible transit service. In this paper, a methodology is proposed to help planners make better decisions regarding the choice between a conventional fixed‐route and a flex‐route policy for a specific transit system with a varying passenger demand. A service quality function is developed to measure the performance of transit systems, and analytical modeling and simulations are used to reproduce transit operation under the two policies. To be closer to reality, two criteria are proposed depending on the processing of rejected requests in the assessment of the service quality function for flex‐route services. In various scenarios, critical demand densities, which represent the switching points between the two competing policies, are derived in a real‐world transit service according to the two criteria. Copyright © 2014 John Wiley & Sons, Ltd.     

Transit network design based on travel time reliability

This paper presents a transit network optimization method, in which travel time reliability on road is considered. A robust optimization model, taking into account the stochastic travel time, is formulated to satisfy the demand of passengers and provide reliable transit service. The optimization model aims to maximize the efficiency of passenger trips in the optimized transit network. Tabu search algorithm is defined and implemented to solve the problem. Then, transit network optimization method proposed in this paper is tested with two numerical examples: a simple route and a medium-size network. The results show the proposed method can effectively improve the reliability of a transit network and reduce the travel time of passengers in general.     

A bi‐level programming approach — Optimal transit fare under line capacity constraints

The fare of a transit line is one of the important decision variables for transit network design. It has been advocated as an efficient means of coordinating the transit passenger flows and of alleviating congestion in the transit network. This paper shows how transit fare can be optimized so as to balance the passenger flow on the transit network and to reduce the overload delays of passengers at transit stops. A bi‐level programming method is developed to optimize the transit fare under line capacity constraints. The upper‐level problem seeks to minimize the total network travel time, while the lower‐level problem is a stochastic user equilibrium transit assignment model with line capacity constraints. A heuristic solution algorithm based on sensitivity analysis is proposed. Numerical example is used to illustrate the application of the proposed model and solution algorithm.     

A stochastic multimodal reliable network design problem under adverse weather conditions

This paper formulates a network design problem (NDP) for finding the optimal public transport service frequencies and link capacity expansions in a multimodal network with consideration of impacts from adverse weather conditions. The proposed NDP aims to minimize the sum of expected total travel time, operational cost of transit services, and construction cost of link capacity expansions under an acceptable level of variance of total travel time. Auto, transit, bus, and walking modes are considered in the multimodal network model for finding the equilibrium flows and travel times. In the proposed network model, demands are assumed to follow Poisson distribution, and weather‐dependent link travel time functions are adopted. A probit‐based stochastic user equilibrium, which is based on the perceived expected travel disutility, is used to determine the multimodal route of the travelers. This model also considers the strategic behavior of the public transport travelers in choosing their routes, that is, common‐line network. Based on the stochastic multimodal model, the mean and variance of total travel time are analytical estimated for setting up the NDP. A sensitivity‐based solution algorithm is proposed for solving the NDP, and two numerical examples are adopted to demonstrate the characteristics of the proposed model. Copyright © 2014 John Wiley & Sons, Ltd.     

A multi‐class mean‐excess traffic equilibrium model with elastic demand

Recent empirical studies have revealed that travel time variability plays an important role in travelers' route choice decisions. To simultaneously account for both reliability and unreliability aspects of travel time variability, the concept of mean‐excess travel time (METT) was recently proposed as a new risk‐averse route choice criterion. In this paper, we extend the mean‐excess traffic equilibrium model to include heterogeneous risk‐aversion attitudes and elastic demand. Specifically, this model explicitly considers (1) multiple user classes with different risk‐aversions toward travel time variability when making route choice decisions under uncertainty and (2) the elasticity of travel demand as a function of METT when making travel choice decisions under uncertainty. This model is thus capable of modeling travelers' heterogeneous risk‐averse behaviors with both travel choice and route choice considerations. The proposed model is formulated as a variational inequality problem and solved via a route‐based algorithm using the modified alternating direction method. Numerical analyses are also provided to illustrate the features of the proposed model and the applicability of the solution algorithm. Copyright © 2012 John Wiley & Sons, Ltd.     

The nonlinear equation system approach to solving dynamic user optimal simultaneous route and departure time choice problems

Dynamic user optimal simultaneous route and departure time choice (DUO-SRDTC) problems are usually formulated as variational inequality (VI) problems whose solution algorithms generally require continuous and monotone route travel cost functions to guarantee convergence. However, the monotonicity of the route travel cost functions cannot be ensured even if the route travel time functions are monotone. In contrast to traditional formulations, this paper formulates a DUO-SRDTC problem (that can have fixed or elastic demand) as a system of nonlinear equations. The system of nonlinear equations is a function of generalized origin-destination (OD) travel costs rather than route flows and includes a dynamic user optimal (DUO) route choice subproblem with perfectly elastic demand and a quadratic programming (QP) subproblem under certain assumptions. This study also proposes a solution method based on the backtracking inexact Broyden–Fletcher–Goldfarb–Shanno (BFGS) method, the extragradient algorithm, and the Frank-Wolfe algorithm. The BFGS method, the extragradient algorithm, and the Frank-Wolfe algorithm are used to solve the system of nonlinear equations, the DUO route choice subproblem, and the QP subproblem, respectively. The proposed formulation and solution method can avoid the requirement of monotonicity of the route travel cost functions to obtain a convergent solution and provide a new approach with which to solve DUO-SRDTC problems. Finally, numeric examples are used to demonstrate the performance of the proposed solution method.     

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.