Dynamic traffic assignment: model classifications and recent advances in travel choice principles

Dynamic Traffic Assignment (DTA) has been studied for more than four decades and numerous reviews of this research area have been conducted. This review focuses on the travel choice principle and the classification of DTA models, and is supplementary to the existing reviews. The implications of the travel choice principle for the existence and uniqueness of DTA solutions are discussed, and the interrelation between the travel choice principle and the traffic flow component is explained using the nonlinear complementarity problem, the variational inequality problem, the mathematical programming problem, and the fixed point problem formulations. This paper also points out that all of the reviewed travel choice principles are extended from those used in static traffic assignment. There are also many classifications of DTA models, in which each classification addresses one aspect of DTA modeling. Finally, some future research directions are identified.postprin

Real-Time Information and Correlations for Optimal Routing in Stochastic Networks

Congestion is a world-wide problem in transportation. One major reason is random interruptions. The traffic network is inherently stochastic, and strong dependencies exist among traffic quantities, e.g., travel time, traffic speed, link volume. Information in stochastic networks can help with adaptive routing in terms of minimizing expected travel time or disutility. Routing in such networks is different from that in deterministic networks or when stochastic dependencies are not taken into account. This dissertation addresses the optimal routing problems, including the optimal a priori path problem and the optimal adaptive routing problem with different information scenarios, in stochastic and time-dependent networks with explicit consideration of the correlations between link travel time random variables. There are a number of studies in the literature addressing the optimal routing problems, but most of them ignore the correlations between link travel times. The consideration of the correlations makes the problem studied in this dissertation difficult, both conceptually and computationally. The optimal path finding problem in such networks is different from that in stochastic and time-dependent networks with no consideration of the correlations. This dissertation firstly provides an empirical study of the correlations between random link travel times and also verifies the importance of the consideration of the spatial and temporal correlations in estimating trip travel time and its reliability. It then shows that Bellman\u27s principle of optimality or non-dominance is not valid due to the time-dependency and the correlations. A new property termed purity is introduced and an exact label-correcting algorithm is designed to solve the problem. With the fast advance of telecommunication technologies, real-time traffic information will soon become an integral part of travelers\u27 route choice decision making. The study of optimal adaptive routing problems is thus timely and of great value. This dissertation studies the problems with a wide variety of information scenarios, including delayed global information, real-time local information, pre-trip global information, no online information, and trajectory information. It is shown that, for the first four partial information scenarios, Bellman\u27s principle of optimality does not hold. A heuristic algorithm is developed and employed based on a set of necessary conditions for optimality. The same algorithm is showed to be exact for the perfect online information scenario. For optimal adaptive routing problem with trajectory information, this dissertation proves that, if the routing policy is defined in a similar way to other four information scenarios, i.e., the trajectory information is included in the state variable, Bellman\u27s principle of optimality is valid. However, this definition results in a prohibitively large number of the states and the computation can hardly be carried out. The dissertation provides a recursive definition for the trajectory-adaptive routing policy, for which the information is not included in the state variable. In this way, the number of states is small, but Bellman\u27s principle of optimality or non-dominance is invalid for a similar reason as in the optimal path problem. Again purity is introduced to the trajectory-adaptive routing policy and an exact algorithm is designed based on the concept of decreasing order of time

Price of Anarchy for Non-atomic Congestion Games with Stochastic Demands

We generalize the notions of user equilibrium and system optimum to non-atomic congestion games with stochastic demands. We establish upper bounds on the price of anarchy for three different settings of link cost functions and demand distributions, namely, (a) affine cost functions and general distributions, (b) polynomial cost functions and general positive-valued distributions, and (c) polynomial cost functions and the normal distributions. All the upper bounds are tight in some special cases, including the case of deterministic demands.Comment: 31 page

Models and Algorithms for Addressing Travel Time Variability: Applications from Optimal Path Finding and Traffic Equilibrium Problems

An optimal path finding problem and a traffic equilibrium problem are two important, fundamental, and interrelated topics in the transportation research field. Under travel time variability, the road networks are considered as stochastic, where the link travel times are treated as random variables with known probability density functions. By considering the effect of travel time variability and corresponding risk-taking behavior of the travelers, this dissertation proposes models and algorithms for addressing travel time variability with applications from optimal path finding and traffic equilibrium problems. Specifically, two new optimal path finding models and two novel traffic equilibrium models are proposed in stochastic networks. To adaptively determine a reliable path with the minimum travel time budget required to meet the user-specified reliability threshold α, an adaptive α-reliable path finding model is proposed. It is formulated as a chance constrained model under a dynamic programming framework. Then, a discrete-time algorithm is developed based on the properties of the proposed model. In addition to accounting for the reliability aspect of travel time variability, the α-reliable mean-excess path finding model further concerns the unreliability aspect of the late trips beyond the travel time budget. It is formulated as a stochastic mixed-integer nonlinear program. To solve this difficult problem, a practical double relaxation procedure is developed. By recognizing travelers are not only interested in saving their travel time but also in reducing their risk of being late, a α-reliable mean-excess traffic equilibrium (METE) model is proposed. Furthermore, a stochastic α-reliable mean-excess traffic equilibrium (SMETE) model is developed by incorporating the travelers’ perception error, where the travelers’ route choice decisions are determined by the perceived distribution of the stochastic travel time. Both models explicitly examine the effects of both reliability and unreliability aspects of travel time variability in a network equilibrium framework. They are both formulated as a variational inequality (VI) problem and solved by a route-based algorithm based on the modified alternating direction method. In conclusion, this study explores the effects of the various aspects (reliability and unreliability) of travel time variability on travelers’ route choice decision process by considering their risk preferences. The proposed models provide novel views of the optimal path finding problem and the traffic equilibrium problem under an uncertain environment, and the proposed solution algorithms enable potential applicability for solving practical problems

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 choos- ing 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 net- work conditions. The model is then tested with GPS data collected in metropolitan Minneapolis-St. Paul, Minnesota. Our data suggest strong correlation among link speed when analyzing morning commute trips. There is no single dominant route (de- fined 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 port- folio of routes could be the rational choice of a traveler who wants to optimize route decisions under variability.Transportation planning, route choice, travel behavior, link performance

Different Policy Objectives of the Road Pricing Problem – a Game Theory Approach

Using game theory we investigate a new approach to formulate and solve optimal tolls with a focus on different policy objectives of the road authority. The aim is to gain more insight into determining optimal tolls as well as into the behavior of users after tolls have been imposed on the network. The problem of determining optimal tolls is stated and defined using utility maximization theory, including elastic demand on the travelers’ side and different objectives for the road authority. Game theory notions are adopted regarding different games and players, rules and outcomes of the games played between travelers on the one hand and the road authority on the other. Different game concepts (Cournot, Stackelberg and monopoly game) are mathematically formulated and the relationship between players, their payoff functions and rules of the games are defined for very simplistic cases. The games are solved for different scenarios and different objectives for the road authority, using the Nash equilibrium concept. Using the Stackelberg game concept as being most realistic for road pricing, a few experiments are presented illustrating the optimal toll design problem subject to different pricing policies considering different objectives of the road authority. Results show different outcomes both in terms of optimal tolls as well as in payoffs for travelers. There exist multiple optimal solutions and objective function may have a non- continuous shape. The main contribution is the two-level separation between of the users from the road authority in terms of their objectives and influences.

INCORPORATING TRAVEL TIME RELIABILITY INTO TRANSPORTATION NETWORK MODELING

Travel time reliability is deemed as one of the most important factors affecting travelers’ route choice decisions. However, existing practices mostly consider average travel time only. This dissertation establishes a methodology framework to overcome such limitation. Semi-standard deviation is first proposed as the measure of reliability to quantify the risk under uncertain conditions on the network. This measure only accounts for travel times that exceed certain pre-specified benchmark, which offers a better behavioral interpretation and theoretical foundation than some currently used measures such as standard deviation and the probability of on-time arrival. Two path finding models are then developed by integrating both average travel time and semi-standard deviation. The single objective model tries to minimize the weighted sum of average travel time and semi-standard deviation, while the multi-objective model treats them as separate objectives and seeks to minimize them simultaneously. The multi-objective formulation is preferred to the single objective model, because it eliminates the need for prior knowledge of reliability ratios. It offers an additional benefit of providing multiple attractive paths for traveler’s further decision making. The sampling based approach using archived travel time data is applied to derive the path semi-standard deviation. The approach provides a nice workaround to the problem that there is no exact solution to analytically derive the measure. Through this process, the correlation structure can be implicitly accounted for while simultaneously avoiding the complicated link travel time distribution fitting and convolution process. Furthermore, the metaheuristic algorithm and stochastic dominance based approach are adapted to solve the proposed models. Both approaches address the issue where classical shortest path algorithms are not applicable due to non-additive semi-standard deviation. However, the stochastic dominance based approach is preferred because it is more computationally efficient and can always find the true optimal paths. In addition to semi-standard deviation, on-time arrival probability and scheduling delay measures are also investigated. Although these three measures share similar mathematical structures, they exhibit different behaviors in response to large deviations from the pre-specified travel time benchmark. Theoretical connections between these measures and the first three stochastic dominance rules are also established. This enables us to incorporate on-time arrival probability and scheduling delay measures into the methodology framework as well

Dynamic route choice in hurricane evacuation

In this research a framework is developed for modeling route choice in hurricane evacuation. Two behavioral hypotheses are evaluated which together with the route choice model, constitute the contributions of the research. The first hypothesis states that beside congestion, other variables such as familiarity with the route, availability of fuel and shelter, facility class, and length of route have an effect on an evacuees\u27 route choice. The second hypothesis states that as time passes and storm conditions change, the impact each variable has on route choice changes. The logit structure was used for modeling the choice process and stated choice data previously collected from the New Orleans area on hypothetical storms was used to calibrate the model. The study found that accessibility of a route, familiarity with a route, facility class, length of a route, and availability of services (gas stations and hotels) had an effect on evacuation route choice. The magnitude of the coefficients of perceived service, accessibility, and distance differed among those evacuating in the first half of the evacuation period versus those that evacuated in the second half but coefficients of facility class were not significantly different between two time intervals. Observed traffic count data from hurricane Katrina evacuation was used to validate the model. Comparison of traffic volumes predicted by the model with actual traffic volumes from hurricane Katrina shows error percentages of 17.5, 0.01, and 28 percent of error for volumes on I-10, I-55, and US-61 respectively

Risk-averse multi-armed bandits and game theory

The multi-armed bandit (MAB) and game theory literature is mainly focused on the expected cumulative reward and the expected payoffs in a game, respectively. In contrast, the rewards and the payoffs are often random variables whose expected values only capture a vague idea of the overall distribution. The focus of this dissertation is to study the fundamental limits of the existing bandits and game theory problems in a risk-averse framework and propose new ideas that address the shortcomings. The author believes that human beings are mostly risk-averse, so studying multi-armed bandits and game theory from the point of view of risk aversion, rather than expected reward/payoff, better captures reality. In this manner, a specific class of multi-armed bandits, called explore-then-commit bandits, and stochastic games are studied in this dissertation, which are based on the notion of Risk-Averse Best Action Decision with Incomplete Information (R-ABADI, Abadi is the maiden name of the author's mother). The goal of the classical multi-armed bandits is to exploit the arm with the maximum score defined as the expected value of the arm reward. Instead, we propose a new definition of score that is derived from the joint distribution of all arm rewards and captures the reward of an arm relative to those of all other arms. We use a similar idea for games and propose a risk-averse R-ABADI equilibrium in game theory that is possibly different from the Nash equilibrium. The payoff distributions are taken into account to derive the risk-averse equilibrium, while the expected payoffs are used to find the Nash equilibrium. The fundamental properties of games, e.g. pure and mixed risk-averse R-ABADI equilibrium and strict dominance, are studied in the new framework and the results are expanded to finite-time games. Furthermore, the stochastic congestion games are studied from a risk-averse perspective and three classes of equilibria are proposed for such games. It is shown by examples that the risk-averse behavior of travelers in a stochastic congestion game can improve the price of anarchy in Pigou and Braess networks. Furthermore, the Braess paradox does not occur to the extent proposed originally when travelers are risk-averse. We also study an online affinity scheduling problem with no prior knowledge of the task arrival rates and processing rates of different task types on different servers. We propose the Blind GB-PANDAS algorithm that utilizes an exploration-exploitation scheme to load balance incoming tasks on servers in an online fashion. We prove that Blind GB-PANDAS is throughput optimal, i.e. it stabilizes the system as long as the task arrival rates are inside the capacity region. The Blind GB-PANDAS algorithm is compared to FCFS, Max-Weight, and c-mu-rule algorithms in terms of average task completion time through simulations, where the same exploration-exploitation approach as Blind GB-PANDAS is used for Max-Weight and c-$\mu$-rule. The extensive simulations show that the Blind GB-PANDAS algorithm conspicuously outperforms the three other algorithms at high loads