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  • AN EMPIRICAL COMPARISON OF ALTERNATIVE USER EQUILIBRIUM TRAFFIC ASSIGNMENT METHODS

    Howard Slavin, Jonathan Brandon, Andres Rabinowicz

    Caliper Corporation

    1. ABSTRACT This paper presents an empirical comparison of alternative methods for computing user equilibrium on large regional transportation networks. Specifically, it examines the solution characteristics, convergence behavior and associated computing times of link-based, origin-based, and path-based methods. The latter two methods use more memory and are computationally more demanding, but have been touted in the literature as converging more rapidly than the link-based Frank-Wolfe method or variants thereof, that are commonly employed in planning software. The principal motivation in searching for improved methods is achieving more rapid and/or tighter convergence in the computation of equilibrium. Greater convergence is needed for accurate forecasting the impacts associated with road and public transport projects and affects nearly all aspects and components of transportation models as well as being a major determinant of their internal consistency. Congested travel speeds are typically used to compute trip distribution and mode choice, and these speeds will be incorrect if a satisfactory traffic assignment is not achieved. Due to long computational times, many models are insufficiently calibrated and converged for forecasting purposes. This problem is partly the result of and is compounded by the slow convergence of the Frank-Wolfe algorithm. In order to perform this research, origin-based and path-based algorithms were coded following the existing literature and tested on networks on large networks of the type that are representative of regional travel demand forecasting efforts for major metropolitan regions. The algorithms coded were those described by Chen et al.(2002) and Dial (2006). An executable version of origin-based traffic assignment code written by one of its proponents (Bar-Gera, 2002) and made freely available for research was also tested. The new algorithms were compared with the current production version of the traffic assignment in TransCAD which is based upon the Frank-Wolfe algorithm and has been shown to be faster than other commercial implementations in the United States. Concurrent with this research, the F-W traffic assignment in TransCAD was multi-threaded leading to speedups in computation proportionate to the number of central processing units and/or CPU cores available. This raises the bar for new assignment algorithms, especially if they are less suitable than F-W for multi-threading.

    ©Association for European Transport and contributors 2006

  • The initial tests revealed that the Bar Gera executable and the Chen et al. path- based method described in the literature would need improvements before they could be competitive in performance computing user equilibrium with existing link-based codes. The path-based method did not converge on medium to large size networks. We made modifications to the gradient search, so that it converged albeit not usually as efficiently as other methods. The Bar Gera origin- based method converged tightly, but only after very long computational times rendering it impractical for use by planners. However, our implementation of an origin user equilibrium method developed in conjunction with Robert Dial and based upon his algorithm B demonstrated superior performance in reaching high levels of convergence in significantly less computing time than F-W. It also can reach a tight equilibrium even more quickly from a prior solution resulting in much lower computing times for models with feedback and most forecasting tasks. Our testing on a realistic planning model trip table and network suggest that the origin user equilibrium can be deployed by practitioners with immediate benefits in terms of reduced computing times and more tightly converged models.

    2. INTRODUCTION This paper reports on an ongoing empirical investigation of alternative traffic assignment methods. Over the last decade, new methods for computing user equilibrium traffic assignments have been researched and have resulted in published claims of superiority over the most widely used methods. Yet these new methods have seen little if any use by practitioners who develop and apply travel demand forecasting models. In this paper, we present an assessment of path-based and origin-based assignment methods and compare their computational performance with the conventional link-based, Frank-Wolfe method in widespread use. The principal motivation in searching for improved methods is achieving more rapid and/or tighter convergence in the computation of equilibrium. Greater convergence is needed for accurate forecasting the impacts associated with road and transit projects and affects nearly all aspects and components of transportation models as well as being a major determinant of their internal consistency. Congested travel speeds are typically used to compute trip distribution and mode choice and these speeds will be incorrect if a satisfactory traffic assignment is not achieved. Due to long computational times, many regional models are insufficiently calibrated and converged for forecasting purposes. This problem is partly the result of and is compounded by the slow convergence of the Frank- Wolfe (FW) algorithm that is used to compute user equilibrium in the most commonly used software packages. In the course of examining the traffic assignment components of many regional models in the U.S. and elsewhere, we identified low levels of convergence, improper measures of convergence, and methods that either do not attempt to calculate user equilibrium or fail to do so correctly. In this paper, we also hope to

    ©Association for European Transport and contributors 2006

  • provide some better guidance on convergence for practicing modelers. We also attempt to provide some information about the nature of the solutions that come from alternative methods. The literature to date has not addressed the properties of these solutions empirically and there are open questions about how similar the solutions are to those currently obtained. During the course of this investigation, computing environments continued to evolve and improve. In particular, inexpensive computers with either two CPUs or two cores or both are now widely available. These computers provide significant speedups for multi-threaded implementations of traffic assignment algorithms, and thus have a bearing upon the comparison of alternative methods. Accordingly, we report results from two generations of hardware. Based upon our findings, the origin user equilibrium approach that implements a version of Dial’s algorithm B can provide superior convergence performance and can be deployed in the near term to achieve reduced computing times, greater convergence, or both. This method has been extended to handle turn penalties and multi-class assignment and will be available commercially in the near future.

    3. BACKGROUND ON USER EQUILIBRIUM In this paper, we focus exclusively on the computation of a static user equilibrium (UE) as defined by Wardrop’s condition that all used paths for each origin- destination pair have the same minimum cost. In other words, no traveler can switch to a shorter path and improve his or her travel time. In congested networks, user equilibrium is characterized by the use of many paths for many O- D pairs. Our focus is on achieving computational rather than representational accuracy. Other traffic assignment models such as stochastic user equilibrium (Sheffi, 1982), bicriterion UE (Dial, 1996; Leurent, 1996), or dynamic models may be behaviorally more realistic. Bernstein (1990) has shown that UE has good stability with respect to small perturbations; consequently, if a tight equilibrium solution can be generated, it should be a computationally stable method of generating forecasts. Results from Boyce et al. (2004) provide empirical support for this conclusion. Achieving much tighter convergence in an equal or lesser amount of computing time would be a breakthrough for practitioners. Beckmann et al. (1956) demonstrated, under assumptions of route costs that are additive link costs and link costs being simply a (continuously differentiable, non- decreasing) function of link flows, that the traffic assignment problem could be formulated as a minimization problem. Leblanc et al. (1975) proposed using the Frank-Wolfe (FW) method for computing equilibrium that was implemented in UTPS and which underlies most planning software implementations in use today. In the FW method, a series of all or nothing assignments are performed and flows are combined using weights derived from a line search that attempts to minimize the linearized objective function. All of the link flows emanating from all

    ©Association for European Transport and contributors 2006

  • origins are updated during each iteration. As a result, the order in which the origins are processed is not of consequence. The process is repeated for a specified number of iterations or until some stopping criterion is met. Note that if the minimum path travel time between each OD pair does not change, the Wardrop condition is satisfied since there are no lower cost alternatives for any traveler. As a result, at equilibrium, the difference between the total cost of the current User Equilibrium, UE, solution ( ) and the total cost of the All-or-Nothing, AON, solution ( ) is zero and the difference is, therefore, a natural measure of convergence. Obtaining the value

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