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.     

International experience with speed limits during and prior to the energy crisis of 1973–4

This paper presents an artificial neural network (ANN) based method for estimating route travel times between individual locations in an urban traffic network. Fast and accurate estimation of route travel times is required by the vehicle routing and scheduling process involved in many fleet vehicle operation systems such as dial‐a‐ride paratransit, school bus, and private delivery services. The methodology developed in this paper assumes that route travel times are time‐dependent and stochastic and their means and standard deviations need to be estimated. Three feed‐forward neural networks are developed to model the travel time behaviour during different time periods of the day‐the AM peak, the PM peak, and the off‐peak. These models are subsequently trained and tested using data simulated on the road network for the City of Edmonton, Alberta. A comparison of the ANN model with a traditional distance‐based model and a shortest path algorithm is then presented. The practical implication of the ANN method is subsequently demonstrated within a dial‐a‐ride paratransit vehicle routing and scheduling problem. The computational results show that the ANN‐based route travel time estimation model is appropriate, with respect to accuracy and speed, for use in real applications.     

Route Choice Models Used in the Stochastic User Equilibrium Problem: A Review

Several route choice models are reviewed in the context of the stochastic user equilibrium problem. The traffic assignment problem has been extensively studied in the literature. Several models were developed focusing mainly on the solution of the link flow pattern for congested urban areas. The behavioural assumption governing route choice, which is the essential part of any traffic assignment model, received relatively much less attention. The core of any traffic assignment method is the route choice model. In the wellknown deterministic case, a simple choice model is assumed in which drivers choose their best route. The assumption of perfect knowledge of travel costs has been long considered inadequate to explain travel behaviour. Consequently, probabilistic route choice models were developed in which drivers were assumed to minimize their perceived costs given a set of routes. The objective of the paper is to review the different route choice models used to solve the traffic assignment problem. Focus is on the different model structures. The paper connects some of the route choice models proposed long ago, such as the logit and probit models, with recently developed models. It discusses several extensions to the simple logit model, as well as the choice set generation problem and the incorporation of the models in the assignment problem.     

A reliability‐based traffic assignment model for multi‐modal transport network under demand uncertainty

In densely populated and congested urban areas, the travel times in congested multi‐modal transport networks are generally varied and stochastic in practice. These stochastic travel times may be raised from day‐to‐day demand fluctuations and would affect travelers' route and mode choice behaviors according to their different expectations of on‐time arrival. In view of these, this paper presents a reliability‐based user equilibrium traffic assignment model for congested multi‐modal transport networks under demand uncertainty. The stochastic bus frequency due to the unstable travel time of bus route is explicitly considered. By the proposed model, travelers' route and mode choice behaviors are intensively explored. In addition, a stochastic state‐augmented multi‐modal transport network is adopted in this paper to effectively model probable transfers and non‐linear fare structures. A numerical example is given to illustrate the merits of the proposed model. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimal speed detector density for the network with travel time information

In the expressway network, detectors are installed on the links for detecting the travel time information while the predicted travel time can be provided by the route guidance system (RGS). The speed detector density can be determined to influence flow distributions in such a way that the precision of the travel time information and the social cost of the speed detectors are optimized, provided that each driver chooses the minimum perceived travel time path in response to the predicted travel time information. In this paper, a bilevel programming model is proposed for the network with travel time information provided by the RGS. The lower-level problem is a probit-based traffic assignment model, while the upper-level problem is to determine the speed detector density that minimizes the measured travel time error variance as well as the social cost of the speed detectors. The sensitivity analysis based algorithm is proposed for the bilevel programming problem. Numerical examples are provided to illustrate the applications of the proposed model and of the solution algorithm.     

A model and solution algorithm for dynamic deterministic user equilibrium assignment

Abstract

In this paper a route-based dynamic deterministic user equilibrium assignment model is presented. Some features of the linear travel time model are first investigated and then a divided linear travel time model is proposed for the estimation of link travel time: it addresses the limitations of the linear travel time model. For the application of the proposed model to general transportation networks, this paper provides thorough investigations on the computational issues in dynamic traffic assignment with many-to-many OD pairs and presents an efficient solution procedure. The numerical calculations demonstrate that the proposed model and solution algorithm produce satisfactory solutions for a network of substantial size with many-to-many OD pairs. Comparisons of assignment results are also made to show the impacts of incorporation of different link travel time models on the assignment results.     

An integrated modelling approach for the bicriterion vehicle routing and scheduling problem with environmental considerations

The consideration of pollution in routing decisions gives rise to a new routing framework where measures of the environmental implications are traded off with business performance measures. To address this type of routing decisions, we formulate and solve a bi-objective time, load and path-dependent vehicle routing problem with time windows (BTL-VRPTW). The proposed formulation incorporates a travel time model representing realistically time varying traffic conditions. A key feature of the problem under consideration is the need to address simultaneously routing and path finding decisions. To cope with the computational burden arising from this property of the problem we propose a network reduction approach. Computational tests on the effect of the network reduction approach on determining non-dominated solutions are reported. A generic solution framework is proposed to address the BTL-VRPTW. The proposed framework combines any technique that creates capacity-feasible routes with a routing and scheduling method that aims to convert the identified routes to problem solutions. We show that transforming a set of routes to BTL-VRPTW solutions is equivalent to solving a bi-objective time dependent shortest path problem on a specially structured graph. We propose a backward label setting technique to solve the emerging problem that takes advantage of the special structure of the graph. The proposed generic solution framework is implemented by integrating the routing and scheduling method into an Ant Colony System algorithm. The accuracy of the proposed algorithm was assessed on the basis of its capability to determine minimum travel time and fuel consumption solutions. Although the computational results are encouraging, there is ample room for future research in algorithmic advances on addressing the proposed problem.     

Modelling rain effects on risk‐taking behaviours of multi‐user classes in road networks with uncertainty

This paper proposes a new travel time reliability‐based traffic assignment model to investigate the rain effects on risk‐taking behaviours of different road users in networks with day‐to‐day demand fluctuations and variations in travel time. A generalized link travel time function is used to capture the rain effects on vehicle travel times and road conditions. This function is further incorporated into daily demand variations to investigate those travel time variations arising from demand uncertainty and rain condition. In view of these rain effects, road users' perception errors on travel times and risk‐taking behaviours on path choices are incorporated in the proposed model with the use of a logit‐based stochastic user equilibrium framework. This new model is formulated as a variational inequality problem in terms of path flows. A numerical example is used to illustrate the application of the proposed model for assessment of the rain effects on road networks with uncertainty.     

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.     

Forecasting journey time distribution with consideration to abnormal traffic conditions

Travel time is an important index for managers to evaluate the performance of transportation systems and an intuitive measure for travelers to choose routes and departure times. An important part of the literature focuses on predicting instantaneous travel time under recurrent traffic conditions to disseminate traffic information. However, accurate travel time prediction is important for assessing the effects of abnormal traffic conditions and helping travelers make reliable travel decisions under such conditions. This study proposes an online travel time prediction model with emphasis on capturing the effects of anomalies. The model divides a path into short links. A Functional Principal Component Analysis (FPCA) framework is adopted to forecast link travel times based on historical data and real-time measurements. Furthermore, a probabilistic nested delay operator is used to calculate path travel time distributions. To ensure that the algorithm is fast enough for online applications, parallel computation architecture is introduced to overcome the computational burden of the FPCA. Finally, a rolling horizon structure is applied to online travel time prediction. Empirical results for Guangzhou Airport Expressway indicate that the proposed method can capture an abrupt change in traffic state and provide a promising and reliable travel time prediction at both the link and path levels. In the case where the original FPCA is modified for parallelization, accuracy and computational effort are evaluated and compared with those of the sequential algorithm. The proposed algorithm is found to require only a piece rather than a large set of traffic incident records.     

Evaluating the benefits of a combined route guidance and road pricing system in a traffic network with recurrent congestion

When drivers do not have complete information on road travel time and thus choose their routes in a stochastic manner or based on their previous experience, separate implementations of either route guidance or road pricing cannot drive a stochastic network flow pattern towards a system optimum in a Wardropian sense. It is thus of interest to consider a combined route guidance and road pricing system. A road guidance system could reduce drivers' uncertainty of travel time through provision of traffic information. A driver who is equipped with a guidance system could be assumed to receive complete information, and hence be able to find the minimum travel time routes in a user-optimal manner, while marginal-cost road pricing could drive a user-optimal flow pattern toward a system optimum. Therefore, a joint implementation of route guidance and road pricing in a network with recurrent congestion could drive a stochastic network flow pattern towards a system optimum, and thus achieve a higher reduction in system travel time. In this paper the interaction between route guidance and road pricing is modeled and the potential benefit of their joint implementation is evaluated based on a mixed equilibrium traffic assignment model. The private and system benefits under marginal-cost pricing and varied levels of market penetration of the information systems are investigated with a small and a large example. It is concluded that the two technologies complement each other and that their joint implementation can reduce travel time more efficiently in a network with recurrent congestion.     

Surrogate‐based optimization of cordon toll levels in congested traffic networks

The benefit, in terms of social surplus, from introducing congestion charging schemes in urban networks is depending on the design of the charging scheme. The literature on optimal design of congestion pricing schemes is to a large extent based on static traffic assignment, which is known for its deficiency in correctly predict travel times in networks with severe congestion. Dynamic traffic assignment can better predict travel times in a road network, but are more computational expensive. Thus, previously developed methods for the static case cannot be applied straightforward. Surrogate‐based optimization is commonly used for optimization problems with expensive‐to‐evaluate objective functions. In this paper, we evaluate the performance of a surrogate‐based optimization method, when the number of pricing schemes, which we can afford to evaluate (because of the computational time), are limited to between 20 and 40. A static traffic assignment model of Stockholm is used for evaluating a large number of different configurations of the surrogate‐based optimization method. Final evaluation is performed with the dynamic traffic assignment tool VisumDUE, coupled with the demand model Regent, for a Stockholm network including 1240 demand zones and 17 000 links. Our results show that the surrogate‐based optimization method can indeed be used for designing a congestion charging scheme, which return a high social surplus. Copyright © 2016 John Wiley & Sons, Ltd.     

Route swapping in dynamic traffic networks

A dynamic traffic assignment (DTA) model typically consists of a traffic performance model and a route choice model. The traffic performance model describes how traffic propagates (over time) along routes connecting origin-destination (OD) pairs, examples being the cell transmission model, the vertical queueing model and the travel time model. This is implemented in a dynamic network loading (DNL) algorithm, which uses the given route inflows to compute the link inflows (and hence link costs), which are then used to compute the route travel times (and hence route costs). A route swap process specifies the route inflows for tomorrow (at the next iteration) based on the route inflows today (at the current iteration). A dynamic user equilibrium (DUE), where each traveller on the network cannot reduce his or her cost of travel by switching to another route, can be sought by iterating between the DNL algorithm and the route swap process. The route swap process itself takes up very little computational time (although route set generation can be very computationally intensive for large networks). However, the choice of route swap process dramatically affects convergence and the speed of convergence. The paper details several route swap processes and considers whether they lead to a convergent system, assuming that the route cost vector is a monotone function of the route inflow vector.     

Dynamic system-optimal traffic assignment using a state space model

We propose a new mathematical formulation for the problem of optimal traffic assignment in dynamic networks with multiple origins and destinations. This problem is motivated by route guidance issues that arise in an Intelligent Vehicle-Highway Systems (IVHS) environment. We assume that the network is subject to known time-varying demands for travel between its origins and destinations during a given time horizon. The objective is to assign the vehicles to links over time so as to minimize the total travel time experienced by all the vehicles using the network. We model the traffic network over the time horizon as a discrete-time dynamical system. The system state at each time instant is defined in a way that, without loss of optimality, avoids complete microscopic detail by grouping vehicles into platoons irrespective of origin node and time of entry to network. Moreover, the formulation contains no explicit path enumeration. The state transition function can model link travel times by either impedance functions, link outflow functions, or by a combination of both. Two versions (with different boundary conditions) of the problem of optimal traffic assignment are studied in the context of this model. These optimization problems are optimal control problems for nonlinear discrete-time dynamical systems, and thus they are amenable to algorithmic solutions based on dynamic programming. The computational challenges associated with the exact solution of these problems are discussed and some heuristics are proposed.     

A portfolio theory of route choice

Although many individual route choice models have been proposed to incorporate travel time variability as a decision factor, they are typically still deterministic in the sense that the optimal strategy requires choosing one particular route that maximizes utility. In contrast, this study introduces an individual route choice model where choosing a portfolio of routes instead of a single route is the best strategy for a rational traveler who cares about both journey time and lateness when facing stochastic network conditions. The proposed model is compared with UE and SUE models and the difference in both behavioral foundation and model characteristics is highlighted. A numerical example is introduced to demonstrate how such model can be used in traffic assignment problem. The model is then tested with GPS data collected in metropolitan Minneapolis–St. Paul, Minnesota. Our data suggest there is no single dominant route (defined here as a route with the shortest travel time for a 15 day period) in 18% of cases when links travel times are correlated. This paper demonstrates that choosing a portfolio of routes could be the rational choice of a traveler who wants to optimize route decisions under variability.     

A path-based flow formulation for the traffic assignment problem

In order to improve cooperation between traffic management and travelers, traffic assignment is the key component to achieve the objectives of both traffic management and route choice decisions for travelers. Traffic assignment can be classified into two models based on the behavioral assumptions governing route choices: User Equilibrium (UE) and System Optimum (SO) traffic assignment. According to UE and SO traffic assignment, travelers usually compete to choose the least cost routes to minimize their own travel costs, while SO traffic assignment requires travelers to work cooperatively to minimize overall cost in the road network. Thus, the paradox of benefits between UE and SO indicates that both are not practical. Thus, a solution technique needs to be proposed to balance UE and SO models, which can compromise both sides and give more feasible traffic assignments. In this paper, Stackelberg game theory is introduced to the traffic assignment problem, which can achieve the trade-off process between traffic management and travelers. Since traditional traffic assignments have low convergence rates, the gradient projection algorithm is proposed to improve efficiency.     

Transport Network Design Problem under Uncertainty: A Review and New Developments

This paper aims to provide a state-of-the-art review of the transport network design problem (NDP) under uncertainty and to present some new developments on a bi-objective-reliable NDP (BORNDP) model that explicitly optimizes the capacity reliability and travel time reliability under demand uncertainty. Both are useful performance measures that can describe the supply-side reliability and demand-side reliability of a road network. A simulation-based multi-objective genetic algorithm solution procedure, which consists of a traffic assignment algorithm, a genetic algorithm, a Pareto filter, and a Monte-Carlo simulation, is developed to solve the proposed BORNDP model. A numerical example based on the capacity enhancement problem is presented to demonstrate the tradeoff between capacity reliability and travel time reliability in the NDP.     

A network equilibrium approach for modelling activity-travel pattern scheduling problems in multi-modal transit networks with uncertainty

An understanding of the interaction between individuals’ activities and travel choice behaviour plays an important role in long-term transit service planning. In this paper, an activity-based network equilibrium model for scheduling daily activity-travel patterns (DATPs) in multi-modal transit networks under uncertainty is presented. In the proposed model, the DATP choice problem is transformed into a static traffic assignment problem by constructing a new super-network platform. With the use of the new super-network platform, individuals’ activity and travel choices such as time and space coordination, activity location, activity sequence and duration, and route/mode choices, can be simultaneously considered. In order to capture the stochastic characteristics of different activities, activity utilities are assumed in this study to be time-dependent and stochastic in relation to the activity types. A concept of DATP budget utility is proposed for modelling the uncertainty of activity utility. An efficient solution algorithm without prior enumeration of DATPs is developed for solving the DATP scheduling problem in multi-modal transit networks. Numerical examples are used to illustrate the application of the proposed model and the solution algorithm.     

Multiclass fuzzy user equilibrium with endogenous membership functions and risk‐taking behaviors

Over the last decades, several approaches have been proposed in the literature to incorporate users' perceptions of travel costs, their bounded rationality, and risk‐taking behaviors into network equilibrium modeling for traffic assignment problem. While theoretically advanced, these models often suffer from high complexity and computational cost and often involve parameters that are difficult to estimate. This study proposes an alternative approach where users' imprecise perceptions of travel times are endogenously constructed as fuzzy sets based on the probability distributions of random link travel times. Two decision rules are proposed accordingly to account for users' heterogeneous risk‐taking behaviors, that is, optimistic and pessimistic rules. The proposed approach, namely, the multiclass fuzzy user equilibrium, can be formulated as a link‐based variational inequality model. The model can be solved efficiently, and parameters involved can be either easily estimated or treated as factors for calibration against observed traffic flow data. Numerical examples show that the proposed model can be solved efficiently even for a large‐scale network of Mashhad, Iran, with 2538 links and 7157 origin–destination pairs. The example also illustrates the calibration capability of the proposed model, highlighting that the model is able to produce much more accurate flow estimates compared with the Wardropian user equilibrium model. Copyright © 2016 John Wiley & Sons, Ltd.     

Stochastic transit equilibrium

We present a transit equilibrium model in which boarding decisions are stochastic. The model incorporates congestion, reflected in higher waiting times at bus stops and increasing in-vehicle travel time. The stochastic behavior of passengers is introduced through a probability for passengers to choose boarding a specific bus of a certain service. The modeling approach generates a stochastic common-lines problem, in which every line has a chance to be chosen by each passenger. The formulation is a generalization of deterministic transit assignment models where passengers are assumed to travel according to shortest hyperpaths. We prove existence of equilibrium in the simplified case of parallel lines (stochastic common-lines problem) and provide a formulation for a more general network problem (stochastic transit equilibrium). The resulting waiting time and network load expressions are validated through simulation. An algorithm to solve the general stochastic transit equilibrium is proposed and applied to a sample network; the algorithm works well and generates consistent results when considering the stochastic nature of the decisions, which motivates the implementation of the methodology on a real-size network case as the next step of this research.