An Empirical Comparison

8/3/2019 An Empirical Comparison

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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 theycould be competitive in performance computing user equilibrium with existinglink-based codes. The path-based method did not converge on medium to largesize networks. We made modifications to the gradient search, so that itconverged albeit not usually as efficiently as other methods. The Bar Gera origin-based method converged tightly, but only after very long computational timesrendering it impractical for use by planners. However, our implementation of an

origin user equilibrium method developed in conjunction with Robert Dial andbased upon his algorithm B demonstrated superior performance in reaching highlevels of convergence in significantly less computing time than F-W.

It also can reach a tight equilibrium even more quickly from a prior solutionresulting in much lower computing times for models with feedback and mostforecasting tasks. Our testing on a realistic planning model trip table andnetwork suggest that the origin user equilibrium can be deployed by practitionerswith immediate benefits in terms of reduced computing times and more tightlyconverged models.

2. INTRODUCTION

This paper reports on an ongoing empirical investigation of alternative trafficassignment methods. Over the last decade, new methods for computing userequilibrium traffic assignments have been researched and have resulted inpublished claims of superiority over the most widely used methods. Yet thesenew methods have seen little if any use by practitioners who develop and applytravel demand forecasting models. In this paper, we present an assessment ofpath-based and origin-based assignment methods and compare theircomputational performance with the conventional link-based, Frank-Wolfe

method in widespread use.

The principal motivation in searching for improved methods is achieving morerapid and/or tighter convergence in the computation of equilibrium. Greaterconvergence is needed for accurate forecasting the impacts associated with roadand transit projects and affects nearly all aspects and components oftransportation models as well as being a major determinant of their internalconsistency. Congested travel speeds are typically used to compute tripdistribution and mode choice and these speeds will be incorrect if a satisfactorytraffic assignment is not achieved. Due to long computational times, manyregional models are insufficiently calibrated and converged for forecasting

purposes. This problem is partly the result of and is compounded by the slowconvergence of the Frank- Wolfe (FW) algorithm that is used to compute userequilibrium in the most commonly used software packages.

In the course of examining the traffic assignment components of many regionalmodels in the U.S. and elsewhere, we identified low levels of convergence,improper measures of convergence, and methods that either do not attempt tocalculate user equilibrium or fail to do so correctly. In this paper, we also hope to

Association for European Transport and contributors 2006


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provide some better guidance on convergence for practicing modelers. We alsoattempt to provide some information about the nature of the solutions that comefrom alternative methods. The literature to date has not addressed the propertiesof these solutions empirically and there are open questions about how similar thesolutions are to those currently obtained.

During the course of this investigation, computing environments continued toevolve and improve. In particular, inexpensive computers with either two CPUs

or two cores or both are now widely available. These computers providesignificant speedups for multi-threaded implementations of traffic assignmentalgorithms, 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 aversion of Dials algorithm B can provide superior convergence performance andcan be deployed in the near term to achieve reduced computing times, greaterconvergence, or both. This method has been extended to handle turn penaltiesand 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 Wardrops condition that all used paths for each origin-destination pair have the same minimum cost. In other words, no traveler canswitch to a shorter path and improve his or her travel time. In congestednetworks, 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 bebehaviorally more realistic. Bernstein (1990) has shown that UE has goodstability with respect to small perturbations; consequently, if a tight equilibriumsolution can be generated, it should be a computationally stable method ofgenerating forecasts. Results from Boyce et al. (2004) provide empirical supportfor this conclusion. Achieving much tighter convergence in an equal or lesseramount of computing time would be a breakthrough for practitioners.

Beckmann et al. (1956) demonstrated, under assumptions of route costs that areadditive link costs and link costs being simply a (continuously differentiable, non-

decreasing) function of link flows, that the traffic assignment problem could beformulated as a minimization problem. Leblanc et al. (1975) proposed using theFrank-Wolfe (FW) method for computing equilibrium that was implemented inUTPS and which underlies most planning software implementations in use today.

In the FW method, a series of all or nothing assignments are performed andflows are combined using weights derived from a line search that attempts tominimize the linearized objective function. All of the link flows emanating from all



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origins are updated during each iteration. As a result, the order in which theorigins are processed is not of consequence. The process is repeated for aspecified number of iterations or until some stopping criterion is met.

Note that if the minimum path travel time between each OD pair does notchange, the Wardrop condition is satisfied since there are no lower costalternatives 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 functionrequires an extra calculation, but the cost at the AON solution (which is alwaysavailable since it determines the direction of search for the next iterate) serves asa lower bound on the equilibrium solution for the current iteration.

UEc

AONc

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

1. Change or percent change of the objective function

n

nn

z

zz1

2. Maximum flow change

n

i

n

i

n

i

x

xx1

max

3. Relative gap

)()()(

UEUE

UEAONUEUE

xcx

xcxxcx

4. Average Excess Cost

OD

xcxxcxUEAONUEUE)()(

where:n Iteration numberi Link index

z Objective functionn

ix Flow at link , at iterationi n

)(c Volume delay function

OD Demand



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The relative gap is the aforementioned difference between the cost of thecurrent UE solution and the cost of the AON solution divided by the cost of thecurrent UE solution. This is a fairly sensitive measure of convergence and issuperior to many other stopping criteria such as simple functions of thedifferences between assignment iterates (Rose et al., 1985). This is the measurethat we rely on for comparisons among alternative methods. In his papers onorigin-based assignment, Bar Gera uses average excess cost which is thedifference between each routes time and the time of the shortest route weighted

by the routes flow (Boyce and Florian, 2005). We also use this measure incomparing the performance of alternative assignment methods. Dial(2006)suggests using a tolerance value that is based upon the difference between thelongest and shortest used paths, and this, too, is probably an effective if not amore effective measure. However, it is not computable for FW solutions so wedo not make use of it in our comparisons.

Apart from these measures, other stopping criteria are often encountered andcan be useful, but some may be potentially misleading. The maximum link flowchange between iterations can help understand the general degree ofconvergence. This measure does not indicate how far from equilibrium a solution

may be. In some planning software, the GAP reported is completely differentfrom the relative gap measure defined above and is computed from successiveUE iterates without consideration of the AON solution. These measures are notcomparable to those employed here and greatly overstate the degree ofconvergence obtained.

Another problem is that some packages do not provide convergent algorithms foruser equilibrium traffic assignment although they purport to do so. Packages thatrely on incremental methods or limit the number of iterations will typically fail toreach 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 isa large literature on how to speed up FW and assignments in general using otherlink based methods such as simplicial decomposition (Patrikson, 1994), there islittle empirical evidence of efficiency gains on the large networks that are used inplanning applications. One exception is the PARTRAN method which reportedlyhas some advantages and is implemented in the EMME/2 package (Florian etal., 1987). The method of successive averages (MSA), which is used to computesome more elaborate traffic assignment models, is generally considered to bemuch slower than FW for congested networks.

The large amount of time required to compute equilibrium on large metropolitannetworks is one reason that regional traffic assignments do not achieve goodconvergence. This has the deepest ramifications for almost all aspects of travelforecasting models. First, congested equilibrium travel times are a keycalibration input for parameter estimation for distribution and mode choicemodels. Second, most models initiate the first round of trip distribution and modesplit model application using congested highway times. If convergence is poor,then trip distribution and mode choice will be affected. Good convergence is also



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needed to make feedback worthwhile and effective. Consequently, many modelsand forecasting applications fail to achieve good convergence and givemisleading 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 andorigin-based methods, both of which require more memory than the FWalgorithm, which is quite efficient in its memory requirements. The rapidadvances in computing power available to demand forecasters and theenormous increases in available memory and storage space make considerationof these methods feasible now when they were not before. Moreover, there is aliterature 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 etal., (2004), Dial (1999), and Dial (2006).

4.1. The Path-based Method

Although path flow solutions are not unique, some proponents erroneouslybelieve that path-based methods are attractive because they lead to a directcharacterization of the routes that are utilized in a traffic assignment solutionrather than just the total link flows. Path based methods require keeping track ofall utilized routes which, for a large problem could be on the order of 100 millionpaths or more. Of course it is just as feasible to save all the paths generated byany other method including F-W and to use compression to store paths moreefficiently. In any case, our focus here is simply on the computational efficiencyof 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 inChen, Lee, and Jayakrishnan (2002). After an initialization with an all-or nothingassignment, the algorithm performs column generation (i.e., it tests for anothershortest path for each O-D pair), and unless terminated because of the stoppingcriterion, it equilibrates the flow over the previously identified paths. Thisequilibration is performed using a gradient search to move flow from longer usedpaths to shorter paths. This method has the advantage that paths can bedropped if they have zero flow. This is more effective than FW which has trouble

moving flow completely away from inefficient paths. The origin-destination pairsare processed one at a time. This is generally thought to be faster because flowand cost updates are relatively inexpensive compared to column generation, butpotentially renders the results order dependent.

The largest network for which Chen et al. reported results was the Advancesketch planning network for Chicago. The machine used was a 256MB RAMSun workstation, and a relative gap of .001 was used as the measure for a



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

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 algorithmB as a path-based algorithm that obviates path storage and enumeration, but inour classification we consider it an origin equilibrium method.

The idea behind origin-based assignment is that the equilibrium solution for eachorigin is an acyclic graph (Jansen and Zozaya-Gorostiza, 1987; Dial, 1999; Bar-Gera, 1999). Origin-based methods maintain acyclic solutions by processing oforigin bushes or subnetworks (Dial, 1999; 2006). This addresses a majorweakness of FW which has trouble removing cycles once they arise, Originapproaches use this subnetwork, have more efficient shortest path calculationsthan FW, and prohibit flow from links that are part of cycles giving greater

computational efficiency. Instead of keeping all the paths in memory, one keepsthe solution for each origin. This multiplies the memory use by a factor relating tothe number of origins although compression can be utilized.

Bar-Geras algorithm has been able to compute solutions to tiny gaps. Hereports average excess costs on the order of 1e -11, this taking more than 2 daysto compute.

When we first noted this, we wondered whether or not this achievement was ofany practical utility. Even if this is overkill, it makes it possible to compareplanning solutions with more highly converged ones.

In any case, Boyce and Bar-Gera deserve credit for calling attention to the poorconvergence associated with most planning models. However, this is notnecessarily an algorithmic problem. It is also related to the use of poor measuresof convergence that give a false sense of security and a lack of appreciation ofconvergence issues on the part of practicing planners. The temptation to controlexecution times has lead to insufficient convergence for the intended purposes ofplanning models.

In reviewing Bar-Geras work, we found the published running times rather highrelative to our experience with UE in existing models. Since machines differ

greatly in terms of performance, we did some joint testing with him andconcluded that while his algorithm might reach low gaps, it was much slower thanour FW implementation for less convergent target gaps. We didnt have accessto the details of Bar-Geras implementation and therefore could not assesswhether or not it could be sped up. Consequently, we focused on implementingan origin-oriented traffic assignment based upon Dials algorithm B.



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5. IMPLEMENTATION

We implemented our own version of the Chen et al. path-based trafficassignment in the C programming language. Our path-based algorithm is thesame as that referenced above, but with a modified gradient search procedure.The modification was necessitated to achieve convergent behavior on the largenetworks that we were using. Examination of the path-based method revealedthat the gradient projection search algorithm was not very efficient and could be

problematic. Consequently, we made some modifications that improved the stepsize calculation. This led to much improved convergence, but it was clear thatmemory requirements and computing times might be too big for the largestnetworks in use.

The Bar Gera source code was not available to us, and the method was notstraightforward to code from the descriptions in papers. Instead, we worked withRobert Dial to implement his algorithm B and created an alternative origin userequilibrium method following his pseudo-code in most respects. Dial (2006) isthe definitive reference for algorithm B. We refer to our implementation as originuser 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 canhandle multiple user classes and turn penalties, both of which are requirementsfor a production code. The initial versions of OUE and our F-W codes weresingle-threaded. Then they were multi-threaded, although not to the samedegree.

Multi-threading is a great way to speed up computations when PCs with multipleCPUs or CPUs with multiple cores are available. However, extra care isnecessary if one wishes to ensure that the results are numerically the same formulti-threaded versus single-threaded implementations. This is important if thesame results are to be obtained when different computers are used. In the case

of the F-W algorithm, multi-threading is relatively straightforward. However, foralgorithms 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 andconverge 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 feedbackloops, it is not uncommon for the effective number of assignments performed tobe multiplied by the number of classes and the number of feedback loops. Thisoften leads to computing times on the order of 25 times or more than that for asingle, single class assignment. This underscores the need for improvedconvergence as well as shortened computational effort.



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We were particularly interested in how much convergence could be achieved inhalf an hour or less which is about the limit (per class) of interest to planners whodo multi-class assignments. Our feeling was that the convergence range ofinterest was a minimum relative gap of .001 or better which is achievable onvirtually all large networks in the U.S in less than one-half hour using a releasedversion of TransCAD on a fast PC in 2005. We were then interested in the abilityto 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 versionwhich is multi-threaded. Consequently, it is twice as fast on a dual core machineand four times as fast on a two CPU machine with two cores per CPU.

We also considered memory utilization which can determine the feasibility ofapplying some of these methods to the largest planning networks currently inuse. Memory requirements are also important because there are definitetradeoffs in execution speed and use of memory. Problems that dont fit inmemory 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 minimalmemory requirements.

We developed a rigorous comparison methodology for performing our tests. Thisinvolved standardizing on computing environments and test networks andadopting consistent measures of effectiveness. For our initial testing, we used a2.0 GHz Pentium M PC with 2 gigabytes of RAM running under Windows XPProfessional. This is a fast single core inexpensive computer. Subsequently weused a 2.2 GHZ dual core Athlon with 2 gigabytes of RAM also running XP

Professional. This machine is quite representative of current offerings, althoughnow during the Summer of 2006 much faster computers are available. Bystandardizing the computing environment, we have minimized the confoundingeffects of memory availability and the speed of hard disk or RAM access canaffect reported running times significantly especially for small networks wherereads and writes might account for a disproportionate amount of total computingtime.

It is harder to control for problem characteristics such as congestion levels andvolume delay functions. Generally speaking, the relationship between travelvolumes and network capacity influences the rate of convergence of all

algorithms. The greater the flow relative to capacity, the slower is theconvergence that will be obtained from the FW algorithm and presumably fromother methods.

Volume delay function properties can also affect convergence. In all of ourcomparisons reported here, we used BPR functions but we experimented withBPR parameters that varied across networks and across the different types oflinks within the same network. However, we cannot be sure that some of our



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conclusions would be different if steeper or flatter volume-delay functions wereutilized.

Networks attributes such as link costs may also differ in their numericalrepresentation even when they supposedly start from the same input values.Numerical treatments range from using integers multiplied by 100, singleprecision floating point calculations, or doubles for arithmetic calculations. Otherpermutations are possible. To avoid undue complications, we used double

precision floating point for computing link flows. Additional precision might bewarranted for super high convergence levels.

Bad test cases can lead to bad conclusions. This fact has been known for a verylong time (Rose et al., 1985). The excessive use of the Sioux Falls and othersmall networks, we believe, has generally been misleading with respect toalgorithmic performance. Bigger networks have many more alternative pathsand much greater computational complexity.

We used a variety of test networks in our research, but we focused most of ourinitial 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, wealso use Bar-Geras Philadelphia network and a network from a model that werecently calibrated for the region surrounding and including Washington, D.C.The latter network was developed by us for a demand modeling project and ishighly accurate in terms of geography. Moreover, the roadway trip table whenassigned matches ground counts closely. The Philadelphia network has just over40,000 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 2523zones.

Concerned that the results still might be problem dependent, we extended ourcomparisons to a variety of other networks in use in the U.S. for majormetropolitan areas. While not reported here, the same general pattern offindings was obtained.

7. MEASURES OF EFFECTIVENESS

The relative gap, the objective function, and average excess costs werecomputed for each traffic assignment solution. Bar-Gera uses average excess

cost which has the same numerator as the relative gap but a different, fixeddenominator which is the number of trips. This measure has a directinterpretation and is potentially more stable because the denominator does notchange as a function of convergence or the number of iterations. We alsocompute the maximum flow change link flow change at different levels ofconvergence. This is the amount that the link flow solution changed on the lastiteration and is an indication of how much of change might be expected if another



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iteration were run. If the maximum flow change is large relative to the potentialimpact 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 cancompute the time it takes to reach this value with rounding or the first iterationthat 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, whichillustrates the rates of convergence for the standard, Frank-Wolfe-basedTransCAD UE and the three alternative assignment programs run on the singlecore 2GHz, Intel Pentium M.

The path-based method has steeper convergence than UE but starts slower andis much slower to reach a relative gap of .0001 than FW. Bar-Geras OBA isslower 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. Itis four times faster than Bar-Geras OBA and about two-thirds of the time takenby OUE. To get to .001, OUE is much faster. It needs less than 10 minuteswhereas UE takes more than 15 minutes. The Bar-Gera executable is especiallyslow 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 toa relative gap of .0001 or better. The path-based method is consistently fasterthan Bar-Geras OBA code.

To reach a relative gap of .0001, the origin user equilibrium assignment based onDials algorithm B needs less than 27 minutes, which we feel is an excellentresult. This is more than three times faster than UE and seven times faster thanOBA.

The excellence convergence of OUE is also illustrated in Figure 2 which plots theobjective 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 maximumflow 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 forthis network and flow matrix, the equivalent values for average excess cost asthe 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 ofAEC.



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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 fastermachine than the one used previously. Figure 3 shows a comparison of theperformance of the standard TransCAD UE on this computer using a single coreand two cores. As is evident, due to multi-threading the dual core solution isobtained 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 orlower. To reach .0001, OUE takes less than one-third the time as Frank-Wolfeand the 21+ minutes taken is quite practical for use. Neither the Bar Gera codenor the path-based codes were explicitly multi-threaded, although there is stillsome 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 .001than for the other methods suggesting that algorithm B is doing more work at thisstage. The small max flow change for UE at .0001 reflects the slow convergenceof Frank-Wolfe.

As illustrated in Table 7, a similar but not identical convergence pattern wasobtained for the Philadelphia network and trip table. While OUE performsexcellently, 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 thatfor Chicago, it appears to be an easier problem to solve. This is reflected in thelesser amount of CPU time required to reach convergence for each method. Our

suspicion is that this result is due to lower congestion in the Philadelphianetwork. Further investigation will be required to understand this result.

The final network tested was, to our way of thinking, the most representative ofnetworks in use today. It comes from a well-calibrated travel demand model thatCaliper developed for the Washington D.C. area for the Maryland-NationalCapital Park and Planning Commission. It has different BPR functions fordifferent types of roadways following Highway Capacity Manual guidelines andfurther 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 isfaster to a relative gap of .001. On a four core machine, FW would be evenfaster, so our conclusion is that OUE is best reserved for tight convergence.

8.2. Differences between FW UE and OUE Link Flows



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Practitioners are notoriously slow to change their modeling methods, and theywill undoubtedly wonder how different the solutions are that generated from theOUE method from those that they currently compute. To address this question,we compared different FW UE solutions for the Washington, D.C. regionalnetwork 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 isa much tighter solution than has ever been used in a deployed forecastingmodel.

Table 13 shows the percent root mean square error (RSME) between the UE andOUE link flows. The first point to note is that highly converged UE and OUE linkflows are similar. This is what we would expect if both algorithms converge tothe same unique equilibrium point as they should. The UE FW result at a .00001gap is reasonably close to the OUE solution at a .0000001 relative gap with anRSME 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 assignmentconverged to a relative gap of .01 or 1 percent is quite far away from a highlyconverged 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 bequite substantial. For this traffic assignment, the maximum link flow difference isnearly 300 trips.

We would not expect that the link flows would be identical at any early pointalong the way to a very tight equilibrium. The reason is that all traffic assignmentsolutions bear the imprint of the algorithm that is employed to compute them.Because of potential order dependence, origin user equilibrium methods are notrecommended 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 forexploring alternative methods. This is that having once obtained a good solutionand saved the results, it should be much faster to compute a new equilibrium fora similar problem.

If this is the case, OUE would reduce the time required when feedback loops arerun. In these situations, the network structure does not change even though thetrip table does.

Normally, when using a F-W based algorithm each traffic assignment takesroughly the same amount of time as the first.

Dial referred to this as a pivot-point assignment. He presented results byfactoring the Chicago trip matrix up and down by various percentages. In allcases the savings were on the order of 50 percent or better to get to a relative



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gap of .0001. We sought to confirm these results on a different network and offera stiffer test to examine the warm start behavior of OUE.

In travel forecasting more generally, changes are made in the network and thereare also changes in the trip table. In theory, OUE may also be able to reduce thetime taken to generate the first traffic assignment for a forecast (when the basecase solution has been run and saved) as well as subsequent ones in feedbackloops.

To examine these issues, we computed a traffic assignment on the regional DCnetwork to a relative gap of .0001 and saved the results. This took 41 minuteson the dual core Athlon. We then randomly increased the number of trips in theorigin-destination matrix by 5 percent. We also doubled the capacities on twosuburban links, reflecting a road improvement. As shown in Table 14, with awarm start from the saved results, OUE was able to compute a new trafficassignment to the same relative gap in 12 minutes. A cold start on the sametraffic 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 fromthe cold start. The computational savings illustrated here are dramatic andsuggest 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 Dials algorithm B offers the prospect of muchtighter convergence than Frank-Wolfe-derived UE traffic assignments and greatlyreduced computing times for very small relative gaps. OUE makes it feasible tocalculate traffic assignments with gaps of .0001 or lower for virtually all largemodels in the U.S. There seems to be little risk in deploying OUE because theresults will be similar to those obtained by current methods if low relative gapsare achieved.

The warm start properties of OUE are particularly attractive for computing modelswith feedback and for forecasting project impacts. In the empirical example weconsidered, 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 generatedby OUE and path-based assignment methods and to provide empirical guidancefor how small the relative gaps should be.

Multithreading is of immediate benefit in accelerating Frank-Wolfe algorithms andwill undoubtedly become vital for other assignment methods as well. Fortunately,



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tighter convergence of user equilibrium models will be more easily achievable inthe 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. Wethank Anthony Chen and Der Hong Lee as well as Shlomo Bekhor for assistancein accessing their research on path-based methods. We would also especiallylike 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-Geras 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-basedassignment, Transportation and Traffic Theory 397-415

Beckman, M., McGuire, C., and Winsten, C. (1956) Studies in the Economicsof 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 TrafficAssignments: How much is enough?, Journal of Transportation Engineering,ASCE, Jan./Feb. 2004

Boyce, D. (2005) Understanding the solution of the sequential procedure withfeedback, unpublished draft paper, January 20, 2005

Boyce, D. and Florian, M. (2005) Workshop on traffic assignment with equilibriummethods, Transportation Research Board Meeting, January 9, 2005

Caliper Corporation (2005) Travel Demand Modeling with TransCAD, Version4.8, Newton, MA

Chen, A., Lee, D-H. and Jayakrishnan, R. (2002) Computational study of state-of-the-art path-based traffic assignment algorithms, Mathematics andComputers in Simulation 59,.509-518



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Dial, R. (1996) Bicriterion Traffic Assignment: Basic theory and elementaryalgorithms, Transportation Science 30/2,.93-111

Dial, R. (1999) Algorithm B: Accurate Traffic Equilibrium (and How to BobtailFrank-Wolfe, Volpe National Transportation Systems Center, Cambridge, MAJuly 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 thePARTRAN variant of the linear approximation method for the networkequilibrium problem, Networks 17,. 319-339

Leurent, F. (1996) The theory and practice of a dual criteria assignment modelwith a continuously distributed value-of-time, in J.B. Lesort (ed) Transportationand Traffic Theory, pp.455-477, Elsevier Science, Ltd. Oxford

Jansen, B. and Zozaya-Gorostiza, C. (1987) The problem of cyclic flows in trafficassignment, Transportation Research, 21B,.299-310

Jayakrishnan, R., Tsai, W.K., Prashker, J., and Rajadhyaksha, S. (1994) A fasterpath-based algorithm for traffic assignment, Transportation Research Record1443. 75-83

LeBlanc, L., Morlok, E., and Pierskalla, W. (1975) An Efficient Approach toSolving 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 convergenceerror in equilibrium traffic assignment models, Transportation Research B, Vol.22B No. 4, 261-274



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FIGURE 1: CONVERGENCE versus CPU TIME - Chicago Network, 2GHzPentiumM

Relative Gap

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

0 50 100 150 200

Time (min)

Gap

O_UE

OBA

FW

Path Based



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

25840000

25860000

25880000

25900000

25920000

25940000

25960000

25980000

26000000

0 10 20 30 40 50 60

Time

ObjectiveFunction

OBA O_UE FW Path Based



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TABLE 1: CPU Minutes to a Relative Gap Chicago Reg. Network, 2GHzPentium 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, 2GHzPentium 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, 2GHzPentium 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



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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 DualAthlon

.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 DualAthlon

.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



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

.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



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TABLE 10: CPU Time (Hours/Minutes/Seconds) to a Rel. Gap D.C. RegionalNetwork, 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 DualAthlon

.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 DualAthlon

.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



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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

GAP HP1

HP2

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



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TABLE 14: Comparison of Warm and Cold Start Assignments on theWashington DC Regional Network

Time to.0001 RG

# Iter. RelativeGap

AEC ObjectiveFunction

MaxFlowDelta

OriginalAssignment

00:41 32 0.000090 0.001673 42477153.37 39.08

IncreaseRand 5% wWarm Start

00:12:10 9 0.000089 0.001718 44935738.00 146.36

IncreaseRand 5%withoutwarm start

00:49:25 39 0.000096 0.001854 44936192.17 23.10

An Empirical Comparison

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