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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


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


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 of the objective function requires an extra calculation, but the cost at the AON solution (which is always available since it determines the direction of search for the next iterate) serves as a lower bound on the equilibrium solution for the current iteration.

UEc

AONc

Since the solution algorithm to the Traffic Assignment Problem is iterative in nature, a stopping criterion is required. Rose et al. (1985) lists several reasonable stopping criteria that might be used and that are shown below:

1. Change or percent change of the objective function

− −

n

nn

zzz 1

2. Maximum flow change

− −

ni

ni

ni

x

xx 1

max

3. Relative gap

∑∑∑⋅

⋅−⋅

)()()(

UEUE

UEAONUEUE

xcxxcxxcx

4. Average Excess Cost

∑∑∑ ⋅−⋅

ODxcxxcx UEAONUEUE )()(

where:

n Iteration number i Link index z Objective function nix Flow at link , at iteration i n)(⋅c Volume delay function

OD Demand


The “relative gap” is the aforementioned difference between the cost of the current UE solution and the cost of the AON solution divided by the cost of the current UE solution. This is a fairly sensitive measure of convergence and is superior to many other stopping criteria such as simple functions of the differences between assignment iterates (Rose et al., 1985). This is the measure that we rely on for comparisons among alternative methods. In his papers on origin-based assignment, Bar Gera uses “average excess cost” which is the difference between each route’s time and the time of the shortest route weighted by the route’s flow (Boyce and Florian, 2005). We also use this measure in comparing the performance of alternative assignment methods. Dial(2006) suggests using a tolerance value that is based upon the difference between the longest and shortest used paths, and this, too, is probably an effective if not a more effective measure. However, it is not computable for FW solutions so we do not make use of it in our comparisons. Apart from these measures, other stopping criteria are often encountered and can be useful, but some may be potentially misleading. The maximum link flow change between iterations can help understand the general degree of convergence. This measure does not indicate how far from equilibrium a solution may be. In some planning software, the “GAP” reported is completely different from the relative gap measure defined above and is computed from successive UE iterates without consideration of the AON solution. These measures are not comparable to those employed here and greatly overstate the degree of convergence obtained. Another problem is that some packages do not provide convergent algorithms for user equilibrium traffic assignment although they purport to do so. Packages that rely on incremental methods or limit the number of iterations will typically fail to reach good solutions. A practical problem for modelers is that the FW algorithm, while efficient early on, exhibits slower convergence as it progresses toward equilibrium. While there is a large literature on how to speed up FW and assignments in general using other link based methods such as simplicial decomposition (Patrikson, 1994), there is little empirical evidence of efficiency gains on the large networks that are used in planning applications. One exception is the PARTRAN method which reportedly has some advantages and is implemented in the EMME/2 package (Florian et al., 1987). The method of successive averages (MSA), which is used to compute some more elaborate traffic assignment models, is generally considered to be much slower than FW for congested networks. The large amount of time required to compute equilibrium on large metropolitan networks is one reason that regional traffic assignments do not achieve good convergence. This has the deepest ramifications for almost all aspects of travel forecasting models. First, congested equilibrium travel times are a key calibration input for parameter estimation for distribution and mode choice models. Second, most models initiate the first round of trip distribution and mode split model application using congested highway times. If convergence is poor, then trip distribution and mode choice will be affected. Good convergence is also


needed to make feedback worthwhile and effective. Consequently, many models and forecasting applications fail to achieve good convergence and give misleading results.

4. ALTERNATIVE ASSIGNMENT METHODS Our motivation in examining alternative methods is to find more rapidly converging algorithms. We consider two classes of methods--path based and origin-based methods, both of which require more memory than the FW algorithm, which is quite efficient in its memory requirements. The rapid advances in computing power available to demand forecasters and the enormous increases in available memory and storage space make consideration of these methods feasible now when they were not before. Moreover, there is a literature that has touted these methods as having superior performance (e.g., Chen et al. (2002); Bar-Gera, (2002); Bar-Gera and Boyce (2002); and Boyce et al., (2004), Dial (1999), and Dial (2006).

4.1. The Path-based Method Although path flow solutions are not unique, some proponents erroneously believe that path-based methods are attractive because they lead to a direct characterization of the routes that are utilized in a traffic assignment solution rather than just the total link flows. Path based methods require keeping track of all utilized routes which, for a large problem could be on the order of 100 million paths or more. Of course it is just as feasible to save all the paths generated by any other method including F-W and to use compression to store paths more efficiently. In any case, our focus here is simply on the computational efficiency of the most effective path based algorithms in computing user equilibrium link flows. The path-based algorithm tested is the method introduced by Jayakrishnan et al. (1994), further evolved by Chen and Jayakrishnan (1998), and described in Chen, Lee, and Jayakrishnan (2002). After an initialization with an all-or nothing assignment, the algorithm performs column generation (i.e., it tests for another shortest path for each O-D pair), and unless terminated because of the stopping criterion, it equilibrates the flow over the previously identified paths. This equilibration is performed using a gradient search to move flow from longer used paths to shorter paths. This method has the advantage that paths can be dropped if they have zero flow. This is more effective than FW which has trouble moving flow completely away from inefficient paths. The origin-destination pairs are processed one at a time. This is generally thought to be faster because flow and cost updates are relatively inexpensive compared to column generation, but potentially renders the results order dependent. The largest network for which Chen et al. reported results was the Advance sketch planning network for Chicago. The machine used was a 256MB RAM Sun workstation, and a relative gap of .001 was used as the measure for a


converged solution. In this research, we tested larger networks and tighter convergence.

4.2. Origin-based Methods Origin-based assignments have been proposed by Bar-Gera (1999) and Dial (1999, 2006). Origin-based methods for UE require more memory than FW, but much less memory than path-based methods. Dial (2006) refers to his algorithm B as a “path-based algorithm that obviates path storage and enumeration,” but in our classification we consider it an origin equilibrium method. The idea behind origin-based assignment is that the equilibrium solution for each origin is an acyclic graph (Jansen and Zozaya-Gorostiza, 1987; Dial, 1999; Bar-Gera, 1999). Origin-based methods maintain acyclic solutions by processing of origin “bushes” or subnetworks (Dial, 1999; 2006). This addresses a major weakness of FW which has trouble removing cycles once they arise, Origin approaches use this subnetwork, have more efficient shortest path calculations than FW, and prohibit flow from links that are part of cycles giving greater computational efficiency. Instead of keeping all the paths in memory, one keeps the solution for each origin. This multiplies the memory use by a factor relating to the number of origins although compression can be utilized. Bar-Gera’s algorithm has been able to compute solutions to tiny gaps. He reports average excess costs on the order of 1e -11, this taking more than 2 days to compute. When we first noted this, we wondered whether or not this achievement was of any practical utility. Even if this is overkill, it makes it possible to compare planning solutions with more highly converged ones. In any case, Boyce and Bar-Gera deserve credit for calling attention to the poor convergence associated with most planning models. However, this is not necessarily an algorithmic problem. It is also related to the use of poor measures of convergence that give a false sense of security and a lack of appreciation of convergence issues on the part of practicing planners. The temptation to control execution times has lead to insufficient convergence for the intended purposes of planning models. In reviewing Bar-Gera’s work, we found the published running times rather high relative to our experience with UE in existing models. Since machines differ greatly in terms of performance, we did some joint testing with him and concluded that while his algorithm might reach low gaps, it was much slower than our FW implementation for less convergent target gaps. We didn’t have access to the details of Bar-Gera’s implementation and therefore could not assess whether or not it could be sped up. Consequently, we focused on implementing an origin-oriented traffic assignment based upon Dial’s algorithm B.


5. IMPLEMENTATION We implemented our own version of the Chen et al. path-based traffic assignment in the C programming language. Our path-based algorithm is the same as that referenced above, but with a modified gradient search procedure. The modification was necessitated to achieve convergent behavior on the large networks that we were using. Examination of the path-based method revealed that the gradient projection search algorithm was not very efficient and could be problematic. Consequently, we made some modifications that improved the step size calculation. This led to much improved convergence, but it was clear that memory requirements and computing times might be too big for the largest networks in use. The Bar Gera source code was not available to us, and the method was not straightforward to code from the descriptions in papers. Instead, we worked with Robert Dial to implement his algorithm B and created an alternative origin user equilibrium method following his pseudo-code in most respects. Dial (2006) is the definitive reference for algorithm B. We refer to our implementation as origin user equilibrium or OUE to distinguish it from Bar-Gera method. After achieving good results with our algorithm B-based OUE, we then created a version that can handle multiple user classes and turn penalties, both of which are requirements for a production code. The initial versions of OUE and our F-W codes were single-threaded. Then they were multi-threaded, although not to the same degree. Multi-threading is a great way to speed up computations when PCs with multiple CPUs or CPUs with multiple cores are available. However, extra care is necessary if one wishes to ensure that the results are numerically the same for multi-threaded versus single-threaded implementations. This is important if the same results are to be obtained when different computers are used. In the case of the F-W algorithm, multi-threading is relatively straightforward. However, for algorithms which are order-dependent such as OUE the same cannot be said. We took the conservative approach and the OUE is less effectively multi-threaded for now.

6. THE TESTING APPROACH To test these methods, we focused on their ability to give good results and converge within a modest amount of computing time. In our experience, algorithms that take a long time to compute are not attractive to practitioners. Given the trend toward multi-class, multi-mode assignments and use of feedback loops, it is not uncommon for the effective number of assignments performed to be multiplied by the number of classes and the number of feedback loops. This often leads to computing times on the order of 25 times or more than that for a single, single class assignment. This underscores the need for improved convergence as well as shortened computational effort.


We were particularly interested in how much convergence could be achieved in half an hour or less which is about the limit (per class) of interest to planners who do multi-class assignments. Our feeling was that the convergence range of interest was a minimum relative gap of .001 or better which is achievable on virtually all large networks in the U.S in less than one-half hour using a released version of TransCAD on a fast PC in 2005. We were then interested in the ability to achieve a relative gap of .0001, the value recommended by Boyce et al. (2004), and which we initially thought might be the lowest value of practical rather than theoretical interest. The traffic assignment used for comparison is the released TransCAD version which is multi-threaded. Consequently, it is twice as fast on a dual core machine and four times as fast on a two CPU machine with two cores per CPU. We also considered memory utilization which can determine the feasibility of applying some of these methods to the largest planning networks currently in use. Memory requirements are also important because there are definite tradeoffs in execution speed and use of memory. Problems that don’t fit in memory can be solved at the expense of execution time associated with reading and writing to disk. Early in the development of traffic assignment methods, minimum memory utilization was a major consideration in designing algorithms, and one of the main advantages of Frank-Wolfe method is that it has minimal memory requirements. We developed a rigorous comparison methodology for performing our tests. This involved standardizing on computing environments and test networks and adopting consistent measures of effectiveness. For our initial testing, we used a 2.0 GHz Pentium M PC with 2 gigabytes of RAM running under Windows XP Professional. This is a fast single core inexpensive computer. Subsequently we used a 2.2 GHZ dual core Athlon with 2 gigabytes of RAM also running XP Professional. This machine is quite representative of current offerings, although now during the Summer of 2006 much faster computers are available. By standardizing the computing environment, we have minimized the confounding effects of memory availability and the speed of hard disk or RAM access can affect reported running times significantly especially for small networks where reads and writes might account for a disproportionate amount of total computing time. It is harder to control for problem characteristics such as congestion levels and volume delay functions. Generally speaking, the relationship between travel volumes and network capacity influences the rate of convergence of all algorithms. The greater the flow relative to capacity, the slower is the convergence that will be obtained from the FW algorithm and presumably from other methods. Volume delay function properties can also affect convergence. In all of our comparisons reported here, we used BPR functions but we experimented with BPR parameters that varied across networks and across the different types of links within the same network. However, we cannot be sure that some of our


conclusions would be different if steeper or flatter volume-delay functions were utilized. Networks attributes such as link costs may also differ in their numerical representation even when they supposedly start from the same input values. Numerical treatments range from using integers multiplied by 100, single precision floating point calculations, or doubles for arithmetic calculations. Other permutations are possible. To avoid undue complications, we used double precision floating point for computing link flows. Additional precision might be warranted for super high convergence levels. Bad test cases can lead to bad conclusions. This fact has been known for a very long time (Rose et al., 1985). The excessive use of the Sioux Falls and other small networks, we believe, has generally been misleading with respect to algorithmic performance. Bigger networks have many more alternative paths and much greater computational complexity. We used a variety of test networks in our research, but we focused most of our initial tests on the large Chicago regional planning network that Bar-Gera used and has made available to researchers. The Chicago network has 39,000+ links, almost 13,000 nodes, and 1771 zones. In the tests reported in this paper, we also use Bar-Gera’s Philadelphia network and a network from a model that we recently calibrated for the region surrounding and including Washington, D.C. The latter network was developed by us for a demand modeling project and is highly accurate in terms of geography. Moreover, the roadway trip table when assigned matches ground counts closely. The Philadelphia network has just over 40,000 links, more than 13,000 nodes, and 1489 zones. The Washington D.C. regional network is much larger and has 62,421 links, 22,286 nodes, and 2523 zones. Concerned that the results still might be problem dependent, we extended our comparisons to a variety of other networks in use in the U.S. for major metropolitan areas. While not reported here, the same general pattern of findings was obtained.

7. MEASURES OF EFFECTIVENESS The relative gap, the objective function, and average excess costs were computed for each traffic assignment solution. Bar-Gera uses average excess cost which has the same numerator as the relative gap but a different, fixed denominator which is the number of trips. This measure has a direct interpretation and is potentially more stable because the denominator does not change as a function of convergence or the number of iterations. We also compute the maximum flow change link flow change at different levels of convergence. This is the amount that the link flow solution changed on the last iteration and is an indication of how much of change might be expected if another


iteration were run. If the maximum flow change is large relative to the potential impact of a project, the forecast may not be at all accurate. There can be a further ambiguity in the use of a specific relative gap. One can compute the time it takes to reach this value with rounding or the first iteration that is strictly below the target value. We use the latter measure.

8. RESULTS The results of our first tests on the Chicago network are shown in Figure 1, which illustrates the rates of convergence for the standard, Frank-Wolfe-based TransCAD UE and the three alternative assignment programs run on the single core 2GHz, Intel Pentium M. The path-based method has steeper convergence than UE but starts slower and is much slower to reach a relative gap of .0001 than FW. Bar-Gera’s OBA is slower still, but the algorithm B-based OUE lived up to its promise and converges rapidly and descends more quickly to lower gaps. As shown in Table 1, UE is the fastest method to obtain a relative gap of .01. It is four times faster than Bar-Gera’s OBA and about two-thirds of the time taken by OUE. To get to .001, OUE is much faster. It needs less than 10 minutes whereas UE takes more than 15 minutes. The Bar-Gera executable is especially slow taking 66 minutes to reach .001 and nearly three hours to reach .0001. On this network, the path-based method is almost competitive with UE but only to a relative gap of .0001 or better. The path-based method is consistently faster than Bar-Gera’s OBA code. To reach a relative gap of .0001, the origin user equilibrium assignment based on Dial’s algorithm B needs less than 27 minutes, which we feel is an excellent result. This is more than three times faster than UE and seven times faster than OBA. The excellence convergence of OUE is also illustrated in Figure 2 which plots the objective function (i.e, the Beckmann value) as a function of CPU time. To get a feel for each level of convergence, Tables 2 and 3 show the maximum flow change and the average excess costs respectively for each method and level of convergence. The results suggest that relative gaps between .001 and .0001 might provide good results for estimating impacts of projects. Note that for this network and flow matrix, the equivalent values for average excess cost as the relative gap are a tougher convergence criterion than the relative gap. Also, note that the AEC measure favors the origin assignment methods at all levels of AEC.


8.1. Multithreaded Results The remainder of the results that will be presented come from runs on the two-core, 2.2 GHZ, AMD dual Athlon. Because it has two cores, this is a faster machine than the one used previously. Figure 3 shows a comparison of the performance of the standard TransCAD UE on this computer using a single core and two cores. As is evident, due to multi-threading the dual core solution is obtained virtually twice as quickly as that for the single core. On the two-core, Dual Athlon, the following results were obtained for Chicago. Recall that while both the UE and the OUE are multi-threaded, the UE is multi-threaded more effectively. As shown in Table 4, to a relative gap of .01, UE (F-W) is the faster algorithm, but OUE becomes better at relative gaps of .001 or lower. To reach .0001, OUE takes less than one-third the time as Frank-Wolfe and the 21+ minutes taken is quite practical for use. Neither the Bar Gera code nor the path-based codes were explicitly multi-threaded, although there is still some gain from the dual core configuration for these methods. An interesting pattern emerges from Table 5. Note that the max flow change at each reported relative gap is progressively larger for OUE above a gap of .001 than for the other methods suggesting that algorithm B is doing more work at this stage. The small max flow change for UE at .0001 reflects the slow convergence of Frank-Wolfe. As illustrated in Table 7, a similar but not identical convergence pattern was obtained for the Philadelphia network and trip table. While OUE performs excellently, the path-based method is the fastest to a relative gap of .0001. Although the traffic assignment problem for Philadelphia is of similar size to that for Chicago, it appears to be an easier problem to solve. This is reflected in the lesser amount of CPU time required to reach convergence for each method. Our suspicion is that this result is due to lower congestion in the Philadelphia network. Further investigation will be required to understand this result. The final network tested was, to our way of thinking, the most representative of networks in use today. It comes from a well-calibrated travel demand model that Caliper developed for the Washington D.C. area for the Maryland-National Capital Park and Planning Commission. It has different BPR functions for different types of roadways following Highway Capacity Manual guidelines and further calibration to traffic counts. As shown in Table 10, on this network, OUE is twice as fast to .0001 as FW and the path-based algorithm is not competitive. However, the multi-threaded FW is faster to a relative gap of .001. On a four core machine, FW would be even faster, so our conclusion is that OUE is best reserved for tight convergence.

8.2. Differences between FW UE and OUE Link Flows


Practitioners are notoriously slow to change their modeling methods, and they will undoubtedly wonder how different the solutions are that generated from the OUE method from those that they currently compute. To address this question, we compared different FW UE solutions for the Washington, D.C. regional network with an OUE solution that was converged to a relative gap of .0000001. The latter computation took slightly more than 5 hours. As far as we know, this is a much tighter solution than has ever been used in a deployed forecasting model. Table 13 shows the percent root mean square error (RSME) between the UE and OUE link flows. The first point to note is that highly converged UE and OUE link flows are similar. This is what we would expect if both algorithms converge to the same unique equilibrium point as they should. The UE FW result at a .00001 gap is reasonably close to the OUE solution at a .0000001 relative gap with an RSME of just 2.8 percent. The maximum difference in link flows is 172 trips. The second point is that the link flow solution to a UE traffic assignment converged to a relative gap of .01 or 1 percent is quite far away from a highly converged solution. Even at a gap of .001 which probably exceeds the tolerance of nearly all large regional models in the U.S., the differences in link flows can be quite substantial. For this traffic assignment, the maximum link flow difference is nearly 300 trips. We would not expect that the link flows would be identical at any early point along the way to a very tight equilibrium. The reason is that all traffic assignment solutions bear the imprint of the algorithm that is employed to compute them. Because of potential order dependence, origin user equilibrium methods are not recommended for traffic assignments unless tight gap tolerances are achieved.

8.3. Warm Start Solution Performance An important benefit cited by Dial (2006) was a key aspect of our motivation for exploring alternative methods. This is that having once obtained a good solution and saved the results, it should be much faster to compute a new equilibrium for a similar problem. If this is the case, OUE would reduce the time required when feedback loops are run. In these situations, the network structure does not change even though the trip table does. Normally, when using a F-W based algorithm each traffic assignment takes roughly the same amount of time as the first. Dial referred to this as a pivot-point assignment. He presented results by factoring the Chicago trip matrix up and down by various percentages. In all cases the savings were on the order of 50 percent or better to get to a relative


gap of .0001. We sought to confirm these results on a different network and offer a stiffer test to examine the warm start behavior of OUE. In travel forecasting more generally, changes are made in the network and there are also changes in the trip table. In theory, OUE may also be able to reduce the time taken to generate the first traffic assignment for a forecast (when the base case solution has been run and saved) as well as subsequent ones in feedback loops. To examine these issues, we computed a traffic assignment on the regional DC network to a relative gap of .0001 and saved the results. This took 41 minutes on the dual core Athlon. We then randomly increased the number of trips in the origin-destination matrix by 5 percent. We also doubled the capacities on two suburban links, reflecting a road improvement. As shown in Table 14, with a warm start from the saved results, OUE was able to compute a new traffic assignment to the same relative gap in 12 minutes. A cold start on the same traffic assignment problem with increased demands took 49 minutes to reach .0001 as this was a harder (i.e., more congested) traffic assignment problem. Also, it is interesting to note that the objective function is lower for the result that begins with the warm start suggesting that it is a better solution than the one from the cold start. The computational savings illustrated here are dramatic and suggest that OUE will confer significant benefits to practitioners.

9. CONCLUSIONS From the empirical comparisons presented, we conclude that the origin user equilibrium method based upon Dial’s algorithm B offers the prospect of much tighter convergence than Frank-Wolfe-derived UE traffic assignments and greatly reduced computing times for very small relative gaps. OUE makes it feasible to calculate traffic assignments with gaps of .0001 or lower for virtually all large models in the U.S. There seems to be little risk in deploying OUE because the results will be similar to those obtained by current methods if low relative gaps are achieved. The warm start properties of OUE are particularly attractive for computing models with feedback and for forecasting project impacts. In the empirical example we considered, using a warm start reduced computing times for traffic assignments to one-fifth of those associated with a regular assignment with a cold start. Further study is warranted to establish the properties of the solutions generated by OUE and path-based assignment methods and to provide empirical guidance for how small the relative gaps should be. Multithreading is of immediate benefit in accelerating Frank-Wolfe algorithms and will undoubtedly become vital for other assignment methods as well. Fortunately,


tighter convergence of user equilibrium models will be more easily achievable in the future.

10. ACKNOWLEDGMENTS The authors are indebted to Robert Dial for his work with us on path and origin-based approaches and for mentoring us on traffic assignment algorithms. We thank Anthony Chen and Der Hong Lee as well as Shlomo Bekhor for assistance in accessing their research on path-based methods. We would also especially like to thank David Boyce and Hillel Bar-Gera for many discussions about origin-based assignments and for providing access to their research and Hillel Bar-Gera’s executable code for OBA.

11. REFERENCES Bar-Gera, H. (2002) Origin-based algorithm for the traffic assignment problem, Transportation Science 36 (4), 398-417 Bar-Gera, H and Boyce, D. (1999) Entropy Maximization in Origin-based assignment, Transportation and Traffic Theory 397-415 Beckman, M., McGuire, C., and Winsten, C. (1956) Studies in the Economics of Transportation, Yale University Press, New Haven Bernstein, D. (1990) Programmability of continuous and discrete network equilibria, Ph.D. thesis, University of Pennsylvania. Boyce, D., Ralevic-Dekic, D., and Bar-Gera, H (2004) Convergence of Traffic Assignments: How much is enough?, Journal of Transportation Engineering, ASCE, Jan./Feb. 2004 Boyce, D. (2005) Understanding the solution of the sequential procedure with feedback, unpublished draft paper, January 20, 2005 Boyce, D. and Florian, M. (2005) Workshop on traffic assignment with equilibrium methods, Transportation Research Board Meeting, January 9, 2005 Caliper Corporation (2005) Travel Demand Modeling with TransCAD, Version 4.8, Newton, MA Chen, A., Lee, D-H. and Jayakrishnan, R. (2002) Computational study of state-of-the-art path-based traffic assignment algorithms, Mathematics and Computers in Simulation 59,.509-518


Dial, R. (1996) Bicriterion Traffic Assignment: Basic theory and elementary algorithms, Transportation Science 30/2,.93-111 Dial, R. (1999) Algorithm B: Accurate Traffic Equilibrium (and How to Bobtail Frank-Wolfe, Volpe National Transportation Systems Center, Cambridge, MA July 25, 1999 Dial, R. (2006) A path-based user-equilibrium traffic assignment algorithm that obviates path storage and enumeration, Transportation Research B, forthcoming Florian, M., Guelat, J., and Spiess, H. (1987), An efficient implementation of the “PARTRAN” variant of the linear approximation method for the network equilibrium problem, Networks 17,. 319-339 Leurent, F. (1996) The theory and practice of a dual criteria assignment model with a continuously distributed value-of-time, in J.B. Lesort (ed) Transportation and Traffic Theory, pp.455-477, Elsevier Science, Ltd. Oxford Jansen, B. and Zozaya-Gorostiza, C. (1987) The problem of cyclic flows in traffic assignment, Transportation Research, 21B,.299-310 Jayakrishnan, R., Tsai, W.K., Prashker, J., and Rajadhyaksha, S. (1994) A faster path-based algorithm for traffic assignment, Transportation Research Record 1443. 75-83 LeBlanc, L., Morlok, E., and Pierskalla, W. (1975) An Efficient Approach to Solving the Road Network Equilibrium Traffic Assignment Problem, Transportation Research Vol 9, 309-318 Patrikson, M. (1994) The Traffic Assignment Problem, VSP, Utrecht Rose, G., Daskin, M., and Koppelman, F. (1988) An examination of convergence error in equilibrium traffic assignment models, Transportation Research B, Vol. 22B No. 4, 261-274


FIGURE 1: CONVERGENCE versus CPU TIME - Chicago Network, 2GHz Pentium M

Relative Gap

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-020 50 100 150 200

Time (min)

Gap

O_UEOBAFWPath Based


FIGURE 2: OBJECTIVE FUNCTION vs. CPU Time- Chicago Net., 2GHz Pentium M

25840000

25860000

25880000

25900000

25920000

25940000

25960000

25980000

26000000

0 10 20 30 40 50 60

Time

Obj

ectiv

e Fu

nctio

n

OBA O_UE FW Path Based


TABLE 1: CPU Minutes to a Relative Gap – Chicago Reg. Network, 2GHz Pentium M

.01 .001 .0001

UE 4.0 15.7 93.9

Path-Based 11.7 34.1 112.0

Bar-Gera OBA 16.0 66.0 175.0

Origin UE 5.9 9.8 26.8

TABLE 2: Max flow change at Relative Gap– Chicago Reg. Network, 2GHz Pentium M

.01 .001 .0001

UE 618.7 40.6 5.4

Path-Based 572.6 81.7 19.5

Bar-Gera OBA NA NA NA

Origin UE 248.9 81.3 9.9

TABLE 3: Average excess costs at Rel. Gap– Chicago Reg. Network, 2GHz Pentium M

.01 .001 .0001

UE 0.23 0.025 0.0025

Path-Based 0.4 0.09 0.01

Bar-Gera OBA 0.14 0.016 0.002

Origin UE 0.15 0.021 0.002


TABLE 4: CPU Time (Hours/Minutes/Seconds) to a Rel. Gap – Chicago Reg. Network, 2.2GHz Dual Athlon

.01 .001 .0001

UE 0:2:43 0:11:32 1:05:25

Path-Based 0:08:43 0:32:21 2:08:08

Bar-Gera OBA 0:12:37 0:52:05 2:14:26

Origin UE 0:04:33 0:10:46 0:21:37

TABLE 5: Max flow change at Rel. Gap– Chicago Reg. Network, 2.2GHz Dual Athlon

.01 .001 .0001

UE 352.44 55.74 3.89

Path-Based 579.22 62.59 23.03

Bar-Gera OBA n/a n/a n/a

Origin UE 119.68 81.89 50.84

TABLE 6: Avg. excess costs at rel. gap– Chicago Reg. Network, 2.2GHz Dual Athlon

.01 .001 .0001

UE 0.2566 0.2512 0.0025

Path-Based 0.196 0.021 0.0025

Bar-Gera OBA 0.2541 0.017 n/a

Origin UE 0.2153 0.0231 0.0024


TABLE 7: CPU Time (Hours/Minutes/Seconds) to a Rel. Gap – Philadelphia, 2.2 GHz Dual Athlon

.01 .001 .0001

UE 0:01:26 0:05:01 0:27:05

Path-Based 0:03:24 0:08:30 0:12:14

Bar-Gera OBA 0:10:51 0:17:24 0:27:06

Origin UE 0:03:05 0:05:27 0:12:45

TABLE 8: Max flow change at relative gap– Philadelphia, 2.2 GHz Dual Athlon

.01 .001 .0001

UE 3596.49 630.10 85.57

Path-Based 8327.75 1857.78 1566.88

Bar-Gera OBA n/a n/a n/a

Origin UE 1873.0 763 139.02

TABLE 9 Average excess costs at relative gap– Philadelphia, 2.2 GHz Dual Athlon

.01 .001 .0001

UE 0.2173 0.022 0.00231

Path-Based 0.1178 0.0073 0.00226

Bar-Gera OBA 0.114 0.048 0.0259

Origin UE 0.11 0.0224 0.0022


TABLE 10: CPU Time (Hours/Minutes/Seconds) to a Rel. Gap – D.C. Regional Network, 2.2 GHz Dual Athlon

.01 .001 .0001

UE 0:04:41 0:19:17 1:21:40

Path-Based 0:48:56 2:44:45 5:22:28

Origin UE 0:11:21 0:23:29 0:42:06

TABLE 11: Max flow change at rel. gap- D.C. Regional Network, 2.2 GHz Dual Athlon

.01 .001 .0001

UE 1554.96 190.89 47.63

Path-Based 3725.12 1265.70 437.69

Origin UE 634.64 194.12 35.35

TABLE 12: Avg. Excess costs at rel. gap- D.C. Regional Network, 2.2 GHz Dual Athlon

.01 .001 .0001

UE 0.1853 0.0175 0.0018

Path-Based 0.1414 0.0099 0.0016

Origin UE 0.1290 0.0148 0.00177


FIGURE 3: Rates of Convergence for One and Two Core Processors- Washington, D.C. Regional Network

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.0045

0.0050

0:00:00 0:15:00 0:30:00 0:45:00 1:00:00 1:15:00 1:30:00

Time

GA

P HP1HP2

TABLE 13: COMPARISON OF UE-FW Link Flows to OUE Link Flows Computed to a 0.000001 Relative Gap– Washington, D.C. Regional Network GAP RMSE Max Flow Difference 0.01 111.214 2719.97 0.001 39.935 2143.85 0.0001 10.15 724.38 0.00001 2.769 171.68



TABLE 14: Comparison of Warm and Cold Start Assignments on the Washington DC Regional Network Time to

.0001 RG # Iter. Relative

Gap AEC Objective

Function Max Flow Delta

Original Assignment

00:41 32 0.000090 0.001673 42477153.37 39.08

Increase Rand 5% w Warm Start

00:12:10 9 0.000089 0.001718 44935738.00 146.36

Increase Rand 5% without warm start

00:49:25 39 0.000096 0.001854 44936192.17 23.10

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