Beyond normality: A cross moment-stochastic user equilibrium model

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Highlights

  • A new stochastic user equilibrium model that accounts for correlation in path utilities is proposed.

  • This is a distributionally robust model that uses mean and covariance information.

  • It is free of distributional assumptions, in particular, does not assume normality.

  • An optimization approach is used to find the traffic equilibrium flow values.

  • Model provides a practical alternative to the probit stochastic user equilibrium.

Abstract

The Stochastic User Equilibrium (SUE) model predicts traffic equilibrium flow assuming that users choose their perceived maximum utility paths (or perceived shortest paths) while accounting for the effects of congestion that arise due to users sharing links. Inspired by recent work on distributionally robust optimization, specifically a Cross Moment (CMM) choice model, we develop a new SUE model that uses the mean and covariance information on path utilities but does not assume the particular form of the distribution. Robustness to distributional assumptions is obtained in this model by minimizing the worst-case expected cost over all distributions with fixed two moments. We show that under mild conditions, the CMM-SUE (Cross Moment-Stochastic User Equilibrium) exists and is unique. By combining a simple projected gradient ascent method to evaluate path choice probabilities with a gradient descent method to find flows, we show that the CMM-SUE is efficiently computable. CMM-SUE provides both modeling flexibility and computational advantages over approaches such as the well-known MNP-SUE (Multinomial Probit-Stochastic User Equilibrium) model that require distributional (normality) assumptions to model correlation effects from overlapping paths. In particular, it avoids the use of simulation methods employed in computations for the distribution-based MNP-SUE model. Preliminary computational results indicate that CMM-SUE provides a practical distributionally robust alternative to MNP-SUE.

Introduction

Traffic assignment problems have intrigued system planners and researchers for several decades. Their key challenge is to develop models and methods that predict traffic flow by incorporating realistic phenomenon such as congestion effects and variations in driver route preferences among others. One of the popular traffic assignment concepts is the notion of a user equilibrium, which reflects the aggregate outcome of individual decisions made by drivers who choose routes from origin to destination nodes on the network. The simplest traffic equilibrium concept is the deterministic Wardropian User Equilibrium (see Wardrop, 1952) which is based on the principle that:

“At equilibrium, the travel costs on all routes that are actually used are equal to or less than those which would be experienced by a user on any unused route.”

Users seek to minimize their own costs of transportation with the traffic flow that results from this principle leading to a User Equilibrium (commonly referred to as UE). At equilibrium, no user can reduce his/her own costs by unilaterally shifting from one route to another. Properties such as the existence of the equilibrium, the uniqueness of the equilibrium, and the efficient computability of the equilibrium have been shown using techniques from convex optimization (see Beckmann et al., 1956), variational inequalities (see Smith, 1979, Dafermos, 1980), and nonlinear complementarity problems (see Aashtiani and Magnanti, 1981).

The Stochastic User Equilibrium (SUE) model introduced by Daganzo and Sheffi (1977) generalizes the deterministic UE model by allowing users to have perceptions of travel costs that are different from the actual realized travel costs. Users choose routes based on their perceived travel costs, thus splitting up the aggregate demand for an origin–destination pair among the various paths connecting them. From a system planner’s view, the perceived travel costs are not observable and hence modeled as random variables with route choice probabilities representing the average fraction of users who move from the origin to the destination along particular paths. The SUE is based on the principle that:

“At equilibrium, no driver can improve on his/her perceived travel cost by unilaterally changing routes.”

In this paper, we employ a convex optimization approach to develop a new SUE model. The main novelty of the model is that the probability distribution of the perceived travel costs is itself uncertain in contrast to traditional SUE models where the probability distribution of the perceived travel costs is fixed. The structure of the paper and the main contributions are as follows:

  • (a)

    In Section 2, we formally introduce the SUE model and provide a literature review on discrete choice models that are commonly used in traffic assignment problems.

  • (b)

    In Section 3, we review the Cross Moment (CMM) choice model that has been recently introduced by Mishra et al., 2012, Ahipasaoglu et al., 2013. In this model, the mean and the covariance matrix of the path utilities are assumed to be known, but the distribution itself is unknown. By focusing on a distribution that maximizes the expected perceived utility, the choice probabilities in the CMM model is computed by solving a convex optimization problem. We provide a simple network with two links for which the choice probabilities are found in closed form for the CMM model and contrast it with the Multinomial Probit (MNP) model. We also discuss a projected gradient ascent algorithm to find the choice probabilities in the CMM model that has been developed by Ahipasaoglu et al. (2013).

  • (c)

    In Section 4, we develop a new stochastic user equilibrium model, referred to as the CMM-SUE model. The CMM-SUE model determines the equilibrium flows as the solution to a distributionally robust optimization problem where the worst-case expected cost in the optimization formulation of Sheffi and Powell, 1982, Daganzo, 1982 is minimized. By accounting for the worst-case distribution, the equilibrium is robust from the system planner’s view. Under mild conditions, we show that the equilibrium exists and is unique. In this section, we also provide a generalization of the logit-based entropy optimization formulation of Fisk (1980) to the CMM-SUE model. As far as we know, no such generalization of Fisk’s optimization formulation is known for the probit model.

  • (d)

    In Section 5, we develop an algorithm to compute the CMM-SUE. The key novelty of the algorithm is in the use of a projected gradient ascent method to compute choice probabilities in contrast to the MNP-SUE model which uses simulation. One advantage of such an approach is that it avoids sampling errors that arise from simulation methods. Furthermore since our algorithm is completely optimization based, to enhance the efficiency in computing the equilibrium we can use the choice probabilities computed in a previous iteration to warm start the optimization problem in the next iteration.

  • (e)

    In Section 6, we present numerical tests to compare the MNP-SUE and CMM-SUE model. We provide evidence ranging from small networks to large networks that indicate the CMM-SUE model effectively captures correlation information among paths while being efficiently computable in comparison to the simulation based MNP-SUE model. The results indicate that CMM-SUE model provides a practical alternative to the MNP-SUE model. Concluding remarks are provided in Section 7.

Section snippets

Notation

Throughout this paper, we use standard letters such as x to denote scalars, bold letters such as x to denote vectors, bold capital letters such as X to denote matrices, the tilde notation such as x̃ to denote random variables, and the italic notation such as X to denote sets (and |X| to denote the size of the set). The n-dimensional Euclidean space is denoted by Rn and the nonnegative orthant by Rn+. The transpose of a column vector xRn is denoted by xT which is a row vector and e is the

Cross moment choice model

Mishra et al., 2012, Ahipasaoglu et al., 2013 have recently introduced a new class of discrete choice models referred to as the Cross Moment (CMM) choice model using the concept of distributionally robustness. In contrast to standard discrete choice models where the distribution θw of the random vector ̃w is completely specified, in distributionally robust models the exact distribution is unknown. Rather the distribution is only known to lie in a set θwΘw. In the CMM model, the set Θw(0,Σw)

Distributionally robust optimization formulation of CMM-SUE

In this section, we develop a new traffic equilibrium model under the assumption that the mean of the random vector ̃w is 0 and the covariance matrix is Σw for wW but the exact distribution is unknown. Our approach is based on the concept of distributionally robust optimization where the objective function is minimized with respect to the worst case distribution from a set of possible distributions. The system planner formulates the distributionally robust counterpart of (2.7) by minimizing

Solution algorithm for CMM-SUE

One of the well-known algorithms to find the arc flows of the MNP-SUE formulation is the Method of Successive Averages (MSA) developed by Sheffi and Powell (1982). MSA is a gradient-based method where in each step the current iterate is updated by adding a multiple of a descent direction. Finding a descent direction is not straightforward because it involves calculating path probabilities. Instead of calculating exact descent directions in each iteration, Sheffi and Powell (1982) proposed to

Numerical tests

In this section, we compare the MNP-SUE and CMM-SUE for five transportation networks discussed in the literature. The first example is a two-link road network for which the CMM-SUE arc flow can be found explicitly. The second example is a five-link road network taken from Daganzo (1982) with linear cost functions while the third example is the classical Braess paradox network taken from Prashker and Bekhor (2004) with nonlinear cost functions. The first three examples have a single

Conclusion

In this paper, we introduce a new stochastic user equilibrium that models the effect of congestion on the distribution of the traffic on the links of a network. The underlying route choice model is based on a recent distributionally robust discrete choice model that is considered in the transportation literature for the first time to the best of our knowledge. The Cross Moment model uses the mean and covariance information for the utilities of different paths without any further distributional

Acknowledgements

The authors would like to thank three reviewers for their detailed comments and valuable suggestions in improving the exposition of the paper. The authors would also like to thank Dongjian Shi for useful discussions on the paper.

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  • Cited by (0)

    This project was partly funded by the SUTD-MIT International Design Center grant number IDG31300105 on ‘Optimization for Complex Discrete Choice’ and the MOE Tier 2 grant number MOE2013-T2-2-168 on ‘Distributional Robust Optimization for Consumer Choice in Transportation Systems’.

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