TRB Planning Applications May 2009, Houston,TX Changing assignment algorithms: the price of better convergence Michael Florian and Shuguang He INRO

TRB Planning Applications May 2009, Houston,TX Contents The need for better convergence New assignment algorithm with paths Results on some test problems Uniqueness considerations Illustration of non unique results Conclusions

TRB Planning Applications May 2008, Houston,TX The need for better convergence -The linear approximation (Frank-Wolfe) algorithm is the most commonly used traffic assignment method -It has the advantage of requiring small amount of RAM -It is easy to explain and is quite robust -It has the drawback of requiring a large number of iterations to obtain a very fine solution -Any path analyses require the re-running of the assignment to obtain the desired results or storing the paths (a very large number) or storing a very large number of paths for a limited number of iterations

TRB Planning Applications May 2008, Houston,TX The need for better convergence -Certain applications require fine solutions in order to compare scenarios and carry out economic evaluation; -The current generation of computers provide plenty of RAM and multiple processors; -This opens up the possibility of implementing faster converging algorithms that require larger amounts of RAM; also it is possible to store and manipulate paths; -It also opens up the possibility of parallel implementations of classical methods.

TRB Planning Applications May 2008, Houston,TX The need for better convergence -The method that we chose to implement is an adaptation of the projected gradient method in the space of path flows; -O-D pairs are considered sequentially with projected gradient descent directions; -It provides finer solutions in much shorter time than that required by the linear approximation method; -Path analyses can be carried out quickly and iterative equilibration algorithms can benefit from the information contained in a previous assignment (warm start)

TRB Planning Applications May 2008, Houston,TX New Equilibrium Traffic Assignment with Path Flows 1.Compute the average cost of all used paths (by O- D pair) 2.Reduce the flow of paths that have a larger cost than the average and 3.Increase the flow on paths that have a smaller cost than the average 4.Just keep the paths with positive flow 5.Add a path if it is shorter than the current equilibrated solution

TRB Planning Applications May 2008, Houston,TX Contents The need for better convergence New assignment algorithm with paths Results on some test problems Uniqueness considerations Illustration of non unique results Conclusions

TRB Planning Applications May 2008, Houston,TX New Equilibrium Traffic Assignment with Path Flows -We compared the performance of the algorithm on several single and multi-class equilibrium assignments -The convergence criterion used for these tests is a measure of Relative Gap (RelGap) for a current iteration: RelGap = Total travel time – Total travel time on shortest path Total travel time - Values of RelGap of the order of 10-5 or 10-6 are considered to be very good

TRB Planning Applications May 2008, Houston,TX Results on some test problems -The platforms used for some of these tests are Dell desktop PC‘s based Intel processors at 2.4 to 3.00 GHz; - Compared algorithms: 1 linear approximation method (Frank-Wolfe) 1000 iterations; 2 projected gradient algorithm.

TRB Planning Applications May 2008, Houston,TX Montreal Regional Planning Network 3 classes 1,425 zones 13,491 nodes 33,540 links

TRB Planning Applications May 2008, Houston,TX Montreal Regional Planning Network 3 classes 1,425 zones 13,491 nodes 33,540 links

Performance Results TRB Planning Applications May 2008, Houston,TX Thanks to Pierre Tremblay, MTQ

Using saved paths for new assignmnet TRB Planning Applications May 2008, Houston,TX Using 2006 assignment for 2015 assignment – about 10% increase in demand ( all to E-6) 2006 assignment min assignment with saved paths min 2015 assignment min Demand Forecast by Mode - MTQ

MAG Regional Planning Network TRB Planning Applications May 2008, Houston,TX 21 modes 2041 centroids regular nodes directional links 1896 turn table entries

MAG Regional Planning Network TRB Planning Applications May 2008, Houston,TX 21 modes 2041 centroids regular nodes directional links 1896 turn table entries

TRB Planning Applications May 2008, Houston,TX 3.12 Ghz – 8 processors at MAG – thanks to Vladimir Livshitz

TRB Planning Applications May 2008, Houston,TX RTA Sydney, Australia Test Network 4 modes 1155 centroids regular nodes directional links 8415 turn table entries

TRB Planning Applications May 2008, Houston,TX RTA Sydney, Australia Test Network 4 modes 1155 centroids regular nodes directional links 8415 turn table entries

TRB Planning Applications May 2008, Houston,TX Performance Results Thanks to Matthew Wilson, RTA

TRB Planning Applications May 2008, Houston,TX Portland Test Network 1,260 zones 8,794 nodes 26,091 links 7,010 turns 4 classes of traffic SOV HOV Heavy Trucks Medium Trucks 2000 Base South Corridor

TRB Planning Applications May 2008, Houston,TX Portland Test Network 1,260 zones 8,794 nodes 26,091 links 7,010 turns 4 classes of traffic SOV HOV Heavy Trucks Medium Trucks 2000 Base South Corridor

TRB Planning Applications May 2008, Houston,TX Performance Results Thanks to Metro Portland

TRB Planning Applications May 2008, Houston,TX SFCTA Test Network 4 classes of traffic 2266 centroids regular nodes directional links 9461 turns

Performance Results TRB Planning Applications May 2008, Houston,TX Thanks to Elisabeth Sall

TRB Planning Applications May 2008, Houston,TX Contents The need for better convergence New assignment algorithm with paths Results on some test problems Uniqueness considerations Illustration of non unique results Conclusions

TRB Planning Applications May 2008, Houston,TX Uniqueness considerations  A little appreciated fact is that the equilibrium assignment does not guarantee unique paths or class flows;  But, running the same code produces the same results so non uniqueness of certain results is a property that was not all that “visible” in practice;  Non uniqueness is a very “elusive” property if one works with the same code.

TRB Planning Applications May 2008, Houston,TX Uniqueness considerations  Different assignment algorithms produce slightly different class flows so results do change; the question is by how much;  Regardless of the algorithm used, the only unique results are the total flows and the class impedances  This remains true if one uses a slightly different implementation of the F&W algorithm so switching F&W implementations would change the results somewhat as well.

TRB Planning Applications May 2008, Houston,TX How different are the results?  The results that may change are all related to the analysis of paths resulting from the assignment;  These include; select link and generalized select link analyses, pure times vs. generalized cost, average tolls paid, sub-area traversal matrices, class flows,…..  Regardless of the algorithm used, the only unique results are the total flows and the class impedances  The implication is that in “feedback” model equilibration one should use schemes that average class impedances and not class volumes!

TRB Planning Applications May 2008, Houston,TX “feedback” equilibration and evaluation  The averaging scheme used should rely on unique results: total link flows or class impedances should be used:  This ensures compatibility with mode and destination choice models and near compatibility with results obtained when the assignment is carried out with the linear approximation method:  Economic evaluation methods based on changes in accessibility times (impedances) will be nearly the same as those obtained with assignments done with the linear approximation method.

TRB Planning Applications May 2008, Houston,TX Contents The need for better convergence New assignment algorithm with paths Results on some test problems Uniqueness considerations Illustration of non unique results Conclusions

TRB Planning Applications May 2008, Houston,TX Chicago Test Network 1790 centroids regular nodes directional links We carried out several select link assignments to see the differences in link flows

TRB Planning Applications May 2008, Houston,TX Select Link Flows Projected gradient flows E-6 Linear aproximation flows E-4 Scale=75

TRB Planning Applications May 2008, Houston,TX Select Link Flow Differences 3 trips Scale=1

TRB Planning Applications May 2008, Houston,TX Select Link Flows Linear approximation flows E-4 Projected gradient flows E-6 Scale=75

TRB Planning Applications May 2008, Houston,TX Select Link Differences 1 trip Scale=1

TRB Planning Applications May 2008, Houston,TX Montreal Regional Planning Network 3 classes 1,425 zones 13,491 nodes 33,540 links We compared the class flows for the Montreal assignment: Linear Approximation at 10^-4 relative gap vs. Projected Gradient at 10^-6 relative gap

TRB Planning Applications May 2008, Houston,TX Montreal network : Total Flows

TRB Planning Applications May 2008, Houston,TX Montreal network : Class 1 SOV

TRB Planning Applications May 2008, Houston,TX Montreal network: Class 2 Light trucks

TRB Planning Applications May 2008, Houston,TX Montreal network: Class 3 heavy trucks

Seattle Regional Planning Model TRB Planning Applications May 2008, Houston,TX 15 modes 30 transit vehicle types 1155 centroids 834 transit lines 5888 regular nodes transit line segments directional links turn table entries

Seattle Regional Planning Model TRB Planning Applications May 2008, Houston,TX 15 modes 30 transit vehicle types 1155 centroids 834 transit lines 5888 regular nodes transit line segments directional links turn table entries

TRB Planning Applications May 2008, Houston,TX Seattle Regional Planning Model - These are the results of comparing the results of the model equilibration after replacing the linear approximation algorithm with the projected gradient algorithm; -The results were provided to us by PSRC staff.

TRB Planning Applications May 2008, Houston,TX PSRC Travel Model Documentation (for Version 1.0)

TRB Planning Applications May 2008, Houston,TX Comparison of Model Equilibration Results 20.4 hrs vs. 10 hrs Results carried out by PSRC planning staff and presented with the permission of PSRC

TRB Planning Applications May 2008, Houston,TX Comparison of Model Equilibration Results 20.4 hrs vs. 10 hrs of computing times (6 “feedback” loops on Intel 2.4 Ghz) Result differences of the order of 0.2% to 0.5%.

TRB Planning Applications May 2008, Houston,TX Comparison of Model Equilibration Results carried out by PSRC planning staff and presented with the permission of PSRC

TRB Planning Applications May 2008, Houston,TX Comparison of Model Equilibration Results 20.4 hrs vs. 10 hrs of computing times (6 “feedback” loops on Intel 2.4 Ghz ) Result differences of the order of 0.2%

Total Flows Comparison Sydney Users' Conference

SOV Impedances Sydney Users' Conference

SOV Travel Time Distribution Sydney Users' Conference

TRB Planning Applications May 2008, Houston,TX Contents The need for better convergence New assignment algorithm with paths Results on some test problems Uniqueness considerations Illustration of non unique results Conclusions

TRB Planning Applications May 2008, Houston,TX It is worth paying the “price” for faster convergence!