Continuous equilibrium network design models

It is known that the network design problem with the assumption of user optimal flows can be modeled as a 0–1 mixed integer programming problem. Instead, we formulate the network design problem with continuous investment variables subject to equilibrium assignment as a nonlinear optimization problem. We show that this optimization problem is equivalent to an unconstrained problem which we solve by direct search techniques. For convex investment cost functions, the performance of both Powell's method and the method of Hooke and Jeeves is approximately the same with respect to computational requirements for a 24 node, 76 arc network. For the case of concave investment functions, Hooke and Jeeves was superior. The solution to the concave continuous model was very similar to that of the 0–1 model. Furthermore, the required solution time was far less than that required by the corresponding discrete model of the same network. The advantages and disadvantages of the continuous approach as well as the computational requirements are discussed.     

Multicriteria evaluation on accessibility‐based transportation equity in road network design problem

This paper investigates the performance of accessibility‐based equity measurements in transportation and proposes a multiobjective optimization model to simulate the trade‐offs between equity maximization and cost minimization of network construction. The equity is defined as the spatial distribution of accessibilities across zone areas. Six representative indicators were formulated, including GINI coefficient, Theil index, mean log deviation, relative mean deviation, coefficient of variation, and Atkinson index, and incorporated into an equity maximization model to evaluate the performance sensitivity. A bilevel multiobjective optimization model was proposed to obtain the Pareto‐optimal solutions for link capacity enhancement in a stochastic road network design problem. A numerical analysis using the Sioux Falls data was implemented. Results verified that the equity indicators are quite sensitive to the pattern of network scenarios in the sense that the level of equity varies according to the amount of overall capacity enhancement as well as the assignment of improved link segments. The suggested multiobjective model that enables representing the Pareto‐optimal solutions can provide multiple options in the decision making of road network design. Copyright © 2012 John Wiley & Sons, Ltd.     

Modeling and solving discrete network design problem with stochastic user equilibrium

In this paper, we address the discrete network design problem, which determines the addition of new roads to existing transportation network to optimize the transportation system performance. Road users are assumed to follow the traffic assignment principle of stochastic user equilibrium. A mixed‐integer nonlinear nonconvex problem is developed to model this discrete network design problem with stochastic user equilibrium. The original problem is relaxed into a convex mixed‐integer nonlinear program, whose solution provides a lower bound of the original problem. The relaxed problem is then embedded into two proposed global optimization solution algorithms to obtain the global optimal solution of the problem. Copyright © 2016 John Wiley & Sons, Ltd.     

Global optimization method for network design problem with stochastic user equilibrium

In this paper, we consider the continuous road network design problem with stochastic user equilibrium constraint that aims to optimize the network performance via road capacity expansion. The network flow pattern is subject to stochastic user equilibrium, specifically, the logit route choice model. The resulting formulation, a nonlinear nonconvex programming problem, is firstly transformed into a nonlinear program with only logarithmic functions as nonlinear terms, for which a tight linear programming relaxation is derived by using an outer-approximation technique. The linear programming relaxation is then embedded within a global optimization solution algorithm based on range reduction technique, and the proposed approach is proved to converge to a global optimum.     

A continuous network design model in stochastic user equilibrium based on sensitivity analysis

The continuous network design problem (CNDP) is known to be difficult to solve due to the intrinsic properties of non‐convexity and nonlinearity. Such kinds of CNDP can be formulated as a bi‐level programme, in which the upper level represents the designer's decisions and the lower level the travellers' responses. Formulations of this kind can be classified as either Stackelberg approaches or Nash ones according to the relationship between the upper level and the lower level parts. This paper formulates the CNDP for road expansion based on Stackelberg game where leader and follower exist, and allows for variety of travellers' behaviour in choosing their routes. In order to solve the problem by the Stackelberg approach, we need a relation between link flows and design parameters. For this purpose, we use a logit route choice model, which provides this in an explicit closed‐form function. This model is applied to two example road networks to test and briefly compare the results between the Stackelberg and Nash approaches to explore the differences between them.     

Speed limits,speed selection and network equilibrium

This paper investigates the local and global impact of speed limits by considering road users’ non-obedient behavior in speed selection. Given a link-specific speed limit scheme, road users will take into account the subjective travel time cost, the perceived crash risk and the perceived ticket risk as determinant factors for their actual speed choice on each link. Homogeneous travelers’ perceived crash risk is positively related to their driving speed. When travelers are heterogeneous, the perceived crash risk is class-specific: different user classes interact with each other and choose their own optimal speed, resulting in a Nash equilibrium speed pattern. With the speed choices on particular roads, travelers make route choices, resulting in user equilibrium in a general network. An algorithm is proposed to solve the user equilibrium problem with heterogeneous users under link-specific speed limits. The models and algorithms are illustrated with numerical examples.     

Approximate network loading and dual-time-scale dynamic user equilibrium

In this paper we present a dual-time-scale formulation of dynamic user equilibrium (DUE) with demand evolution. Our formulation belongs to the problem class that Pang and Stewart (2008) refer to as differential variational inequalities. It combines the within-day time scale for which route and departure time choices fluctuate in continuous time with the day-to-day time scale for which demand evolves in discrete time steps. Our formulation is consistent with the often told story that drivers adjust their travel demands at the end of every day based on their congestion experience during one or more previous days. We show that analysis of the within-day assignment model is tremendously simplified by expressing dynamic user equilibrium as a differential variational inequality. We also show there is a class of day-to-day demand growth models that allow the dual-time-scale formulation to be decomposed by time-stepping to yield a sequence of continuous time, single-day, dynamic user equilibrium problems. To solve the single-day DUE problems arising during time-stepping, it is necessary to repeatedly solve a dynamic network loading problem. We observe that the network loading phase of DUE computation generally constitutes a differential algebraic equation (DAE) system, and we show that the DAE system for network loading based on the link delay model (LDM) of Friesz et al. (1993) may be approximated by a system of ordinary differential equations (ODEs). That system of ODEs, as we demonstrate, may be efficiently solved using traditional numerical methods for such problems. To compute an actual dynamic user equilibrium, we introduce a continuous time fixed-point algorithm and prove its convergence for effective path delay operators that allow a limited type of nonmonotone path delay. We show that our DUE algorithm is compatible with network loading based on the LDM and the cell transmission model (CTM) due to Daganzo (1995). We provide a numerical example based on the much studied Sioux Falls network.     

A note of network equilibrium and noncooperative games

Rosenthal has shown that a user-optimized transportation network is equivalent to a pure Strategy Nash equilibrium when the network flows are discrete. Noting that most network equilibrium theorists take flows to be continuous, we extend this result to the nondiscrete case. We prove that a continuous flow, user-optimized network is a pure-strategy Nash equilibrium in a game with a continuum of pure strategies. Our “game”, however, differs from Rosenthal's in its players, strategies, and payoffs. For instance, the players in our model are not the motorists, but the origin-destination pairs. Some possible applications and extensions of our results are discussed.     

The corridor problem: Preliminary results on the no-toll equilibrium

Consider a traffic corridor that connects a continuum of residential locations to a point central business district, and that is subject to flow congestion. The population density function along the corridor is exogenous, and except for location vehicles are identical. All vehicles travel along the corridor from home to work in the morning rush hour, and have the same work start-time but may arrive early. The two components of costs are travel time costs and schedule delay (time early) costs. Determining equilibrium and optimum traffic flow patterns for this continuous model, and possible extensions, is termed “The Corridor Problem”. Equilibria must satisfy the trip-timing condition, that at each location no vehicle can experience a lower trip price by departing at a different time. This paper investigates the no-toll equilibrium of the basic Corridor Problem.     

Electric vehicle charging station locations: Elastic demand,station congestion,and network equilibrium

Battery-only electric vehicles (BEVs) generally offer better air quality through lowered emissions, along with energy savings and security. The issue of long-duration battery charging makes charging-station placement and design key for BEV adoption rates. This work uses genetic algorithms to identify profit-maximizing station placement and design details, with applications that reflect the costs of installing, operating, and maintaining service equipment, including land acquisition. Fast electric vehicle charging stations (EVCSs) are placed across a congested city's network subject to stochastic demand for charging under a user-equilibrium traffic assignment. BEV users’ station choices consider endogenously determined travel times and on-site charging queues. The model allows for congested-travel and congested-station feedback into travelers’ route choices under elastic demand and BEV owners’ station choices, as well as charging price elasticity for BEV charging users.Boston-network results suggest that EVCSs should locate mostly along major highways, which may be a common finding for other metro settings. If 10% of current EV owners seek to charge en route, a user fee of $6 for a 30-min charging session is not enough for station profitability under a 5-year time horizon in this region. However, $10 per BEV charging delivers a 5-year profit of $0.82 million, and 11 cords across 3 stations are enough to accommodate a near-term charging demand in this Boston-area application. Shorter charging sessions, higher fees, and/or allowing for more cords per site also increase profits generally, everything else constant. Power-grid and station upgrades should keep pace with demand, to maximize profits over time, and avoid on-site congestion.     

Preferences for travel time under risk and ambiguity: Implications in path selection and network equilibrium

In this paper, we study the preferences for uncertain travel times in which probability distributions may not be fully characterized. In evaluating an uncertain travel time, we explicitly distinguish between risk, where the probability distribution is precisely known, and ambiguity, where it is not. In particular, we propose a new criterion called ambiguity-aware CARA travel time (ACT) for evaluating uncertain travel times under various attitudes of risk and ambiguity, which is a preference based on blending the Hurwicz criterion and Constant Absolute Risk Aversion (CARA). More importantly, we show that when the uncertain link travel times are independently distributed, finding the path that minimizes travel time under the ACT criterion is essentially a shortest path problem. We also study the implications on Network Equilibrium (NE) model where travelers on the traffic network are characterized by their knowledge of the network uncertainty as well as their risk and ambiguity attitudes under the ACT. We derive and analyze the existence and uniqueness of solutions under NE. Finally, we obtain the Price of Anarchy that characterizes the inefficiency of this new equilibrium. The computational study suggests that as uncertainty increases, the influence of selfishness on inefficiency diminishes.     

Service network design with asset management: Formulations and comparative analyses

In this paper, we address the service network design with asset management problem, which integrates asset management considerations into service network design models for consolidation-based freight carriers. We propose model formulations based on arc variables for both flow and design, as well as formulations with path flow variables and new cycle design variables. Problem instances reflecting actual planning problems are used in the computational study to analyze the strengths and weaknesses of the various model formulations and the impact of asset management considerations on the transportation plan and the computational effort. Experimental results indicate that formulations based on cycle variables outperform traditional arc-based formulations, and that considering asset management issues may significantly impact the outcome of service planning models.     

Intermodal hub-and-spoke network design: Incorporating multiple stakeholders and multi-type containers

This paper develops a mathematical program with equilibrium constraints (MPEC) model for the intermodal hub-and-spoke network design (IHSND) problem with multiple stakeholders and multi-type containers. The model incorporates a parametric variational inequality (VI) that formulates the user equilibrium (UE) behavior of intermodal operators in route choice for any given network design decision of the network planner. The model also uses a cost function that is capable of reflecting the transition from scale economies to scale diseconomies in distinct flow regimes for carriers or hub operators, and a disutility function integrating actual transportation charges and congestion impacts for intermodal operators. To solve the MPEC model, a hybrid genetic algorithm (HGA) embedded with a diagonalization method for solving the parametric VI is proposed. Finally, the comparative analysis of the HGA and an exhaustive enumeration algorithm indicates a good performance of the HGA in terms of computational time and solution quality. The HGA is also applied to solve a large-scale problem to show the applicability of the proposed model and algorithm.     

The effect of the price of gasoline on the urban economy: From route choice to general equilibrium

RELU-TRAN2, a spatial computable general equilibrium (CGE) model of the Chicago MSA is used to understand how gasoline use, car-VMT, on-the-road fuel intensity, trips and location patterns, housing, labor and product markets respond to a gas price increase. We find a long-run elasticity of gasoline demand (with congestion endogenous) of ?0.081, keeping constant car prices and the TFI (technological fuel intensity) of car types but allowing consumers to choose from car types. 43% of this long run elasticity is from switching to transit; 15% from trip, car-type and location choice; 38% from price, wage and rent equilibration, and 4% from building stock changes. 79% of the long run elasticity is from changes in car-VMT (the extensive margin) and 21% from savings in gasoline per mile (the intensive margin); with 83% of this intensive margin from changes in congestion and 17% from the substitution in favor of lower TFI. An exogenous trend-line improvement of the TFI of the car-types available for choice raises the long-run response to a percent increase in the gas price from ?0.081 to ?0.251. Thus, only 1/3 of the long-run response to the gas price stems from consumer choices and 2/3 from progress in fuel intensity. From 2000 to 2007, real gas prices rose 53.7%, the average car fuel intensity improved 2.7% and car prices fell 20%. The model predicts that from these changes alone, keeping constant population, income, etc. aggregate gasoline use in this period would have fallen by 5.2%.     

Pareto-improving transportation network design and ownership regimes

Private provision of public roads signifies co-existence of free, public-tolled and private-tolled roads. This paper investigates the Pareto-improving transportation network design problem under various ownership regimes by allowing joint choice of road pricing and capacity enhancement on free links. The problem of interest is formulated as a bi-objective mathematical programming model that considers the travel cost of road users in each origin-destination pair and the investment return of the whole network. The non-dominated Pareto-improving solutions of toll and/or capacity enhancement schemes are sought for achieving a win-win situation. A sufficient condition is provided for the existence of the non-dominated Pareto-improving schemes and then the properties of those schemes are analyzed. It is found that, under some mild assumptions, the optimal capacity enhancement is uniquely determined by the link flow under any non-dominated Pareto-improving scheme. As a result, the joint road pricing and capacity enhancement problem reduces to a bi-objective second-best road pricing problem. A revenue distribution mechanism with return rate guarantee is proposed to implement the non-dominated Pareto-improving schemes.     

Validation and use of equilibrium network assignment for urban highway reconstruction planning

The need to rehabilitate interstate highways and bridges will increase tremendously in the next decades. Due to traffic restrictions imposed during construction, these rehabilitation activities will cause major disruptions in existing traffic patterns. In order to develop mitigation strategies to reduce such travel impacts, reliable forecasts of likely travel pattern changes would be beneficial. In this paper, we examine the suitability of using an equilibrium traffic assignment model to predict the impacts of a major highway reconstruction project. A case study of travel impacts during reconstruction of I-376, the Parkway East, in Pittsburgh, Pennsylvania is made to validate the adequacy of the network assignment model. Results are compared with actual volume counts collected during periods with and without traffic restrictions. The model produced estimates of link volumes that were, on average, from 16% to 28% different from the observed link counts along two screenlines. Large discrepancies with some of the counts could be explained in part by aberrations in the observed data or in the network model's structure. A sketch-planning analysis is also performed, and the results are compared with those from the network assignment model. The network assignment model is also used to predict the impacts of a hypothetical reconstruction scenario in which the Parkway East is totally closed during its reconstruction.     

Dynamic network equilibrium for daily activity-trip chains of heterogeneous travelers: application to large-scale networks

Applications of dynamic network equilibrium models have, mostly, considered the unit of traffic demand either as one-way trip, or as multiple independent trips. However, individuals’ travel patterns typically follow a sequence of trips chained together. In this study we aim at developing a general simulation-based dynamic network equilibrium algorithm for assignment of activity-trip chain demand. The trip chain of each individual trip maker is defined by the departure time at origin, sequence of activity destination locations, including the location of their intermediate destinations and their final destination, and activity duration at each of the intermediate destinations. Spatial and temporal dependency of subsequent trips on each other necessitate time and memory consuming calculations and storage of node-to-node time-dependent least generalized cost path trees, which is not practical for very large metropolitan area networks. We first propose a reformulation of the trip-based demand gap function formulation for the variational inequality formulation of the Bi-criterion Dynamic User Equilibrium (BDUE) problem. Next, we propose a solution algorithm for solving the BDUE problem with daily chain of activity-trips. Implementation of the algorithm for very large networks circumvents the need to store memory-intensive node-to-node time-dependent shortest path trees by implementing a destination-based time-dependent least generalized cost path finding algorithm, while maintaining the spatial and temporal dependency of subsequent trips. Numerical results for a real-world large scale network suggest that recognizing the dependency of multiple trips of a chain, and maintaining the departure time consistency of subsequent trips provide sharper drops in gap values, hence, the convergence could be achieved faster (compared to when trips are considered independent of each other).     

Dynamic user equilibrium with side constraints for a traffic network: Theoretical development and numerical solution algorithm

This paper investigates a traffic volume control scheme for a dynamic traffic network model which aims to ensure that traffic volumes on specified links do not exceed preferred levels. The problem is formulated as a dynamic user equilibrium problem with side constraints (DUE-SC) in which the side constraints represent the restrictions on the traffic volumes. Travelers choose their departure times and routes to minimize their generalized travel costs, which include early/late arrival penalties. An infinite-dimensional variational inequality (VI) is formulated to model the DUE-SC. Based on this VI formulation, we establish an existence result for the DUE-SC by showing that the VI admits at least one solution. To analyze the necessary condition for the DUE-SC, we restate the VI as an equivalent optimal control problem. The Lagrange multipliers associated with the side constraints as derived from the optimality condition of the DUE-SC provide the traffic volume control scheme. The control scheme can be interpreted as additional travel delays (either tolls or access delays) imposed upon drivers for using the controlled links. This additional delay term derived from the Lagrange multiplier is compared with its counterpart in a static user equilibrium assignment model. If the side constraint is chosen as the storage capacity of a link, the additional delay can be viewed as the effort needed to prevent the link from spillback. Under this circumstance, it is found that the flow is incompressible when the link traffic volume is equal to its storage capacity. An algorithm based on Euler’s discretization scheme and nonlinear programming is proposed to solve the DUE-SC. Numerical examples are presented to illustrate the mechanism of the proposed traffic volume control scheme.     

Consumer preferences relative to the price and network capability of small urban vehicles

Preferences of consumers for small urban vehicle concepts differing only with respect to their hypothetical purchase prices and network capabilities (i.e., whether they are capable of operating on expressways, major arterials or local streets) are analyzed using statistical techniques based on psychological scaling theories. Results from these analyses indicate that a vast majority of consumers are not readily willing to give up the accessibility provided by conventional automobiles. More specifically, over the range of hypothetical prices considered here, network capability dominates as a determinant of preferences for vehicle concepts. Also, the ability to operate vehicles on expressways is of utmost importance to consumers.