Bounded rationality and irreversible network change

A network change is said to be irreversible if the initial network equilibrium cannot be restored by revoking the change. The phenomenon of irreversible network change has been observed in reality. To model this phenomenon, we develop a day-to-day dynamic model whose fixed point is a boundedly rational user equilibrium (BRUE) flow. Our BRUE based approach to modeling irreversible network change has two advantages over other methods based on Wardrop user equilibrium (UE) or stochastic user equilibrium (SUE). First, the existence of multiple network equilibria is necessary for modeling irreversible network change. Unlike UE or SUE, the BRUE multiple equilibria do not rely on non-separable link cost functions, which makes our model applicable to real-world large-scale networks, where well-calibrated non-separable link cost functions are generally not available. Second, travelers’ boundedly rational behavior in route choice is explicitly considered in our model. The proposed model is applied to the Twin Cities network to model the flow evolution during the collapse and reopening of the I-35 W Bridge. The results show that our model can to a reasonable level reproduce the observed phenomenon of irreversible network change.     

Global convergence of the trial-and-error method for the traffic-restraint congestion-pricing scheme with day-to-day flow dynamics

The traffic-restraint congestion-pricing scheme (TRCPS) aims to maintain traffic flow within a desirable threshold for some target links by levying the appropriate link tolls. In this study, we propose a trial-and-error method using observed link flows to implement the TRCPS with the day-to-day flow dynamics. Without resorting to the origin–destination (O–D) demand functions, link travel time functions and value of time (VOT), the proposed trial-and-error method works as follows: tolls for the traffic-restraint links are first implemented each time (trial) and they are subsequently updated using observed link flows in a disequilibrium state at any arbitrary time interval. The trial-and-error method has the practical significance because it is necessary only to observe traffic flows on those tolled links and it does not require to wait for the network flow pattern achieving the user equilibrium (UE) state. The global convergence of the trial-and-error method is rigorously demonstrated under mild conditions. We theoretically show the viability of the proposed trial-and-error method, and numerical experiments are conducted to evaluate its performance. The result of this study, without doubt, enhances the confidence of practitioners to adopt this method.     

Boundedly rational user equilibria (BRUE): Mathematical formulation and solution sets

Boundedly rational user equilibria (BRUE) represent traffic flow distribution patterns where travellers can take any route whose travel cost is within an ‘indifference band’ of the shortest path cost. Those traffic flow patterns satisfying the above condition constitute a set, named the BRUE solution set. It is important to obtain all the BRUE flow patterns, because it can help predict the variation of the link flow pattern in a traffic network under the boundedly rational behavior assumption. However, the methodology of constructing the BRUE set has been lacking in the established literature. This paper fills the gap by constructing the BRUE solution set on traffic networks with fixed demands. After defining ε-BRUE, where ε is the indifference band for the perceived travel cost, we formulate the ε-BRUE problem as a nonlinear complementarity problem (NCP), so that a BRUE solution can be obtained by solving a BRUE–NCP formulation. To obtain the BRUE solution set encompassing all BRUE flow patterns, we propose a methodology of generating acceptable path set which may be utilized under the boundedly rational behavior assumption. We show that with the increase of the indifference band, the acceptable path set that contains boundedly rational equilibrium flows will be augmented, and the critical values of indifference band to augment these path sets can be identified by solving a family of mathematical programs with equilibrium constraints (MPEC) sequentially. The BRUE solution set can then be obtained by assigning all traffic demands to the acceptable path set. Various numerical examples are given to illustrate our findings.     

Second best toll pricing within the framework of bounded rationality

The network design problem is usually formulated as a bi-level program, assuming the user equilibrium is attained in the lower level program. Given boundedly rational route choice behavior, the lower-level program is replaced with the boundedly rational user equilibria (BRUE). The network design problem with boundedly rational route choice behavior is understudied due to non-uniqueness of the BRUE. In this study, thus, we mainly focus on boundedly rational toll pricing (BR-TP) with affine link cost functions. The topological properties of the lower level BRUE set are first explored. As the BRUE solution is generally non-unique, urban planners cannot predict exactly which equilibrium flow pattern the transportation network will operate after a planning strategy is implemented. Due to the risk caused by uncertainty of people’s reaction, two extreme scenarios are considered: the traffic flow patterns with either the minimum system travel cost or the maximum, which is the “risk-prone” (BR-TP-RP) or the “risk-averse” (BR-TP-RA) scenario respectively. The upper level BR-TP is to find an optimal toll minimizing the total system travel cost, while the lower level is to find the best or the worst scenario. Accordingly BR-TP can be formulated as either a min –min or a min –max program. Solution existence is discussed based on the topological properties of the BRUE and algorithms are proposed. Two examples are accompanied to illustrate the proposed methodology.     

A discrete dynamical system of formulating traffic assignment: Revisiting Smith’s model

Through relaxing the behavior assumption adopted in Smith’s model (Smith, 1984), we propose a discrete dynamical system to formulate the day-to-day evolution process of traffic flows from a non-equilibrium state to an equilibrium state. Depending on certain preconditions, the equilibrium state can be equivalent to a Wardrop user equilibrium (UE), Logit-based stochastic user equilibrium (SUE), or boundedly rational user equilibrium (BRUE). These equivalence properties indicate that, to make day-to-day flows evolve to equilibrium flows, it is not necessary for travelers to choose their routes based on actual travel costs of the previous day. Day-to-day flows can still evolve to equilibrium flows provided that travelers choose their routes based on estimated travel costs which satisfy these preconditions. We also show that, under a more general assumption than the monotonicity of route cost function, the trajectory of the dynamical system converges to a set of equilibrium flows by reasonably setting these parameters in the dynamical system. Finally, numerical examples are presented to demonstrate the application and properties of the dynamical system. The study is helpful for understanding various processes of forming traffic jam and designing an algorithm for calculating equilibrium flows.     

Network reliability‐based optimal toll design

Optimal toll design from a network reliability point of view is addressed in this paper. Improving network reliability is proposed as a policy objective of road pricing. A reliability‐based optimal toll design model, where on the upper level network performance including travel time reliability is optimized, while on the lower level a dynamic user‐equilibrium is achieved, is presented. Road authorities aim to optimize network travel time reliability by setting tolls in a network design problem. Travelers are influenced by these tolls and make route and trip decisions by considering travel times and tolls. Network performance reliability is analyzed for a degradable network with elastic and fluctuated travel demand, which integrates reliability and uncertainty, dynamic network equilibrium models, and Monte Carlo methods. The proposed model is applied to a small hypothesized network for which optimal tolls are derived. The network travel time reliability is indeed improved after implementing optimal tolling system. Trips may have a somewhat higher, but more reliable, travel time.     

The impact of speed limits on traffic equilibrium and system performance in networks

Speed limits are usually imposed on roads in an attempt to enhance safety and sometimes serve the purpose of reducing fuel consumption and vehicular emissions as well. Most previous studies up to date focus on investigation of the effects of speed limits from a local perspective, while network-wide traffic reallocation effects are overlooked. This paper makes the first attempt to investigate how a link-specific speed limit law reallocates traffic flow in an equilibrium manner at a macroscopic network level. We find that, although the link travel time–flow relationship is altered after a speed limit is imposed, the standard traffic assignment method still applies. With the commonly adopted assumptions, the uniqueness of link travel times at user equilibrium (UE) remains valid, and the UE flows on links with non-binding speed limits are still unique. The UE flows on other links with binding speed limits may not be unique but can be explicitly characterized by a polyhedron or a linear system of equalities and inequalities. Furthermore, taking into account the traffic reallocation effects of speed limits, we compare the capability of speed limits and road pricing for decentralizing desirable network flow patterns. Although from a different perspective for regulating traffic flows with a different mechanism, a speed limit law may play the same role as a toll charge scheme and perform better than some negative (rebate) toll schemes under certain conditions for network flow management.     

Global optimization methods for the discrete network design problem

This paper addresses the discrete network design problem (DNDP) with multiple capacity levels, or multi-capacity DNDP for short, which determines the optimal number of lanes to add to each candidate link in a road network. We formulate the problem as a bi-level programming model, where the upper level aims to minimize the total travel time via adding new lanes to candidate links and the lower level is a traditional Wardrop user equilibrium (UE) problem. We propose two global optimization methods by taking advantage of the relationship between UE and system optimal (SO) traffic assignment principles. The first method, termed as SO-relaxation, exploits the property that an optimal network design solution under SO principle can be a good approximate solution under UE principle, and successively sorts the solutions in the order of increasing total travel time under SO principle. Optimality is guaranteed when the lower bound of the total travel time of the unexplored solutions under UE principle is not less than the total travel time of a known solution under UE principle. The second method, termed as UE-reduction, adds the objective function of the Beckmann-McGuire-Winsten transformation of UE traffic assignment to the constraints of the SO-relaxation formulation of the multi-capacity DNDP. This constraint is convex and strengthens the SO-relaxation formulation. We also develop a dynamic outer-approximation scheme to make use of the state-of-the-art mixed-integer linear programming solvers to solve the SO-relaxation formulation. Numerical experiments based on a two-link network and the Sioux-Falls network are conducted.     

Dynamic congestion pricing with day-to-day flow evolution and user heterogeneity

This paper investigates evolutionary implementation of congestion pricing schemes to minimize the system cost and time, measured in monetary and time units, respectively, with the travelers’ day-to-day route adjustment behavior and their heterogeneity. The travelers’ heterogeneity is captured by their value-of-times. First, the multi-class flow dynamical system is proposed to model the travelers’ route adjustment behavior in a tolled transportation network with multiple user classes. Then, the stability condition and properties of equilibrium is examined. We further investigate the trajectory control problem via dynamic congestion pricing scheme to derive the system cost, time optimum, and generally, Pareto optimum in the sense of simultaneous minimization of system cost and time. The trajectory control problem is modeled by a differential–algebraic system with the differential sub-system capturing the flow dynamics and the algebraic one capturing the pricing constraint. The explicit Runge–Kutta method is proposed to calculate the dynamic flow trajectories and anonymous link tolls. The method allows the link tolls to be updated with any predetermined periods and forces the system cost and/or time to approach the optimum levels. Both analytical and numerical examples are adopted to examine the efficiency of the method.     

Network-wide adaptive tolling for connected and automated vehicles

This article proposes Δ-tolling, a simple adaptive pricing scheme which only requires travel time observations and two tuning parameters. These tolls are applied throughout a road network, and can be updated as frequently as travel time observations are made. Notably, Δ-tolling does not require any details of the traffic flow or travel demand models other than travel time observations, rendering it easy to apply in real-time. The flexibility of this tolling scheme is demonstrated in three specific traffic modeling contexts with varying traffic flow and user behavior assumptions: a day-to-day pricing model using static network equilibrium with link delay functions; a within-day adaptive pricing model using the cell transmission model and dynamic routing of vehicles; and a microsimulation of reservation-based intersection control for connected and autonomous vehicles with myopic routing. In all cases, Δ-tolling produces significant benefits over the no-toll case, measured in terms of average travel time and social welfare, while only requiring two parameters to be tuned. Some optimality results are also given for the special case of the static network equilibrium model with BPR-style delay functions.     

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.     

Experiment of boundedly rational route choice behavior and the model under satisficing rule

In this paper, we study the boundedly rational route choice behavior under the Simon’s satisficing rule. A laboratory experiment was carried out to verify the participants’ boundedly rational route choice behavior. By introducing the concept of aspiration level which is specific to each person, we develop a novel model of the problem in a parallel-link network and investigate the properties of the boundedly rational user equilibrium (BRUE) state. Conditions for ensuring the existence and uniqueness of the BRUE solution are derived. A solution method is proposed to find the unique BRUE state. Extensions to general networks are conducted. Numerical examples are presented to demonstrate the theoretical analyses.     

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.     

Multi-class percentile user equilibrium with flow-dependent stochasticity

Travelers often reserve a buffer time for trips sensitive to arrival time in order to hedge against the uncertainties in a transportation system. To model the effects of such behavior, travelers are assumed to choose routes to minimize the percentile travel time, i.e. the travel time budget that ensures their preferred probability of on-time arrival; in doing so, they drive the system to a percentile user equilibrium (UE), which can be viewed as an extension of the classic Wardrop equilibrium. The stochasticity in the supply of transportation are incorporated by modeling the service flow rate of each road segment as a random variable. Such stochasticity is flow-dependent in the sense that the probability density functions of these random variables, from which the distribution of link travel time are constructed, are specified endogenously with flow-dependent parameters. The percentile route travel time, obtained by directly convolving the link travel time distributions in this paper, is not available in closed form in general and has to be numerically evaluated. To reveal their structural properties, percentile UE solutions are examined in special cases and verified with numerical results. For the general multi-class percentile UE traffic assignment problem, a variational inequality formulation is given and solved using a route-based algorithm. The algorithm makes use of the diagonal elements in the Jacobian of percentile route travel time, which is approximated through recursive convolution. Preliminary numerical experiments indicate that the algorithm is able to achieve highly precise equilibrium solutions.     

On the degradation of performance for traffic networks with oblivious users

We consider the problem of characterizing user equilibria and optimal solutions for routing in a given network. We extend the known models by considering users oblivious to congestion in the following sense: While in the typical user equilibrium setting the users follow a strategy that minimizes their individual cost by taking into account the (dynamic) congestion due to the current routing pattern, an oblivious user ignores congestion altogether; instead, he or she decides his routing on the basis of cheapest routes on a network without any flow whatsoever. These cheapest routes can be, for example, the shortest paths in the network without any flow. This model tries to capture the fact that a certain percentage of travelers base their route simply on the distances they observe on a map, without thinking (or knowing, or caring) about the delays experienced on this route due to their fellow travelers. In this work we study the effect of such users using as the measure of network performance its price of anarchy, i.e., the ratio of the total latency experienced by the users (oblivious or not) at equilibrium over the social optimum.     

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.     

A link-node complementarity model and solution algorithm for dynamic user equilibria with exact flow propagations

In this paper, we propose a link-node complementarity model for the basic deterministic dynamic user equilibrium (DUE) problem with single-user-class and fixed demands. The model complements link-path formulations that have been widely studied for dynamic user equilibria. Under various dynamic network constraints, especially the exact flow propagation constraints, we show that the continuous-time dynamic user equilibrium problem can be formulated as an infinite dimensional mixed complementarity model. The continuous-time model can be further discretized as a finite dimensional non-linear complementarity problem (NCP). The proposed discrete-time model captures the exact flow propagation constraints that were usually approximated in previous studies. By associating link inflow at the beginning of a time interval to travel times at the end of the interval, the resulting discrete-time model is predictive rather than reactive. The solution existence and compactness condition for the proposed model is established under mild assumptions. The model is solved by an iterative algorithm with a relaxed NCP solved at each iteration. Numerical examples are provided to illustrate the proposed model and solution approach. We particularly show why predictive DUE is preferable to reactive DUE from an algorithmic perspective.     

Dynamic user optimal traffic assignment on congested multidestination networks

An equivalent continuous time optimal control problem is formulated to predict the temporal evolution of traffic flow pattern on a congested multiple origin-destination network, corresponding to a dynamic generalization of Wardropian user equilibrium. Optimality conditions are derived using the Pontryagin minimum principle and given economic interpretations, which are generalizations of similar results previously reported for single-destination networks. Analyses of sufficient conditions for optimality and of singular controls are also given. Under the steady-state assumptions, the model is shown to be a proper dynamic extension of Beckmann's mathematical programming problem for a static user equilibrium traffic assignment.     

Effect of advanced traveler information systems and road pricing in a network with non-recurrent congestion

The effect of the application of advanced transport information system (ATIS) and road pricing is studied in a transportation system under non-recurrent congestion. A stochastic network deterministic user equilibrium model (SNDUE) with elastic demand is formulated and used to evaluate the welfare and private impacts of different market penetrations of ATIS, together with road pricing for a simple network. Both marginal first-best road pricing and a second-best fixed road pricing are considered. The incentives of private users to use ATIS are analyzed and the characteristics of optimum tolls as a function of ATIS market penetration are shown. We conclude that ATIS is an efficient and necessary tool to reduce the effects of non-recurrent incidents in a transportation network, especially when non-recurrent congestion causes a significant deterioration of operational conditions of the network. If the impact of non-recurrent incidents on free flow costs is small or is reduced only to congestion effects, the use of road pricing would be more efficient. Social benefits obtained when jointly implementing ATIS and road pricing are practically the same whether first-best or second-best road pricing is used. Considering the private costs perceived by the network users, and the benefits experienced by equipped users, the maximum level of market penetration achieved could be limited because private benefits disappear after certain market penetration is obtained.     

Learning marginal-cost pricing via a trial-and-error procedure with day-to-day flow dynamics

This paper investigates the convergence of the trial-and-error procedure to achieve the system optimum by incorporating the day-to-day evolution of traffic flows. The path flows are assumed to follow an ‘excess travel cost dynamics’ and evolve from disequilibrium states to the equilibrium day by day. This implies that the observed link flow pattern during the trial-and-error procedure is in disequilibrium. By making certain assumptions on the flow evolution dynamics, we prove that the trial-and-error procedure is capable of learning the system optimum link tolls without requiring explicit knowledge of the demand functions and flow evolution mechanism. A methodology is developed for updating the toll charges and choosing the inter-trial periods to ensure convergence of the iterative approach towards the system optimum. Numerical examples are given in support of the theoretical findings.